A theoretical device in historical linguistics that I think can easily go abused is the basic morphophonological concept of “alternation”.
To lay some groundwork: an initial issue, on which I may expand more at some point, is that several grades of what is meant by “alternation” in the first place can be distinguished. All of them come with their own behavior, and trying to treat them as equal is a surefire way of going off-track in analysis.
Firstly, some archetypal examples of morphophonological alternation are easy to think of: systematic phenomena like consonant gradation in Finnic and Samic (and the less-known case of Nganasan), or consonant mutation in Celtic. These permeate a language’s lexicon on all levels, including neologisms and other newly gained vocabulary, and are often employed for specific grammatical functions.
It would be an obvious error to treat all morphophonology as having similar wide-reaching signifigance though. In what I would call the second category, even indubitably productive alternation patterns can be far more minor, applying perhaps only to a single morpheme. Consider e.g. the voicing and vocalization alternations in the English past tense morpheme -(e)d and the plural morpheme -(e)s; these require separate accounts to cover all corner cases (zero-suffix pasts like led, trod and found are not quite the same as zero-suffix plurals like fish, sheep and mice), and what similarities they show in their phonological behavior are easily seen to result from general phonetic constraints — not from them sharing abstract “suffix mutation rules”.
Thirdly it is quite common for particular types of alternations to be at least partly lexicalized. Examples like English teach : taught (< *tǣk-ja- : *tǣx-t) or Finnish niellä ‘to swallow’ ~ nälkä ‘hunger’ (< *ńälə- ~ *ńäl-kä) have obviously long since ceased to be anything but fossilized relicts. This may quite well go also for “marginally productive” alternations that only replicate themselves by analogy. Nobody would claim that PIE ablaut is productive in English, and regardless this has not stopped people from creating new strong past tense forms like shit : shat (given here the analogy of sit : sat).
Unproductive alternations can also go deeper yet, involving loaning. English is a good source of examples: we can consider e.g. Germanic/Romance doublets, such as stand ~ statue (going all the way back to PIE), or for a slightly younger example, ward ~ guard (the latter originally a Frankish loan in French, and thus linked at the West Germanic level). Cases where both sides are of loan origin are possible as well, e.g. Latin/Greek doublets such as serpent ~ herpetology, Latin/French doublets such as regal ~ royal, or doublets originally derived within Latin, such as cause ~ excuse (← causa ~ *ex-causa). Such alternations might be impossible to identify at sight, and only a deeper knowledge of etymology and language history will end up demonstrating that they in fact go back to a common root. 
(There would be also a neurolinguistics blog post to be written on if “productive morpho(phono)logy”, as separate from phonology, (morpho)syntax and lexicon exists as its own phenomenon at all, or if it’s all simply an issue of more or less fossilized analogies — but that’s not my main topic, nor really even particularly within my expertise.)
There is additionally a fourth sense of “alternation” however, which I think goes the least appreciated: language-internal false cognates. Whenever alternations of some sort occur within the paradigm of a single word, it’s usually a good starting idea to suppose some kind of a historical divergence, rather than flat-out suppletion. Whenever two words aren’t directly related though, and only show some degree of semantic and phonetic resemblance, presuming a relationship is far more risky. A comparison of e.g. English beak with peak, though perhaps plausible on the face, does not suffice to allow us to infer the existence of “an alternation b ~ p” — except in the banal descriptive sense that these two semantically close-by words really do differ only in their initial consonant. Historically, this similarity appears to be entirely accidental.
I’ve avoided giving too many examples above to steer clear of feeding confirmation bias. While languages typically indeed contain numerous unproductive doublets and marginal alternations, these can be entirely indistinguishable from mere chance similarities. I would consider it methodologically invalid to claim that just because two words show similarity, they should be considered etymologically related through “some” unspecified means. This kind of a conclusion should always require specific confirmation from other comparative data.
Not necessarily language-external comparison, mind you. E.g. if an alternation can be attributed to a sound change in a particular context, it would be expected that the same change has affected other words as well, and therefore created multiple similar doublets. For a specific example, lexical doublets in Finnish of the type sortaa ‘to break down; to oppress’ ~ sorsia ‘to tease’ (the latter appearing to be derived from the former with the frequentative suffix -i-) can be put on a much firmer ground already as soon as compared with the existence of an alternation -t- : -si- also in inflection, as in e.g. kaartaa ‘to move in an arc, to go around’ : past tense kaarsi. 
Aside from “pure” chance similarity, another risk involved in doublet-hunting is semantic contamination: similar shape can lead two words to drift toward similar meanings, if given the chance. One cautionary example could be Finnish kastua ‘to become wet’ (also kastaa ‘to dip in’, kaste ‘dew’ etc.) ~ kostua ‘to become moist’ (also kostea ‘moist’), which at first sight appear to be some kind of a related doublet, perhaps comparable to other examples of an “a ~ o alternation” (say, kajo ‘shimmer’ ~ koi ‘dawn’ )?
However, we know from historical, dialect and other Finnic data that the original meaning of kostua has been ‘to return, to be returned’ (and compare in Modern Finnish still the expression kostua jostain ‘to benefit from (a scheme or deal)’ ) — which allows it to be regularly analyzed as a reflexive derivative of the base verb kostaa, whose main meaning nowadays is ‘to avenge’ (< ‘*to return something’), clearly unrelated to moisture or wetness. The meaning ‘to become moist’ seems to have developed through the stage ‘to return to usable condition’, which for traditional leather-based (and, to some extent, wooden) tools and items can well have meant remoisturization after drying. Also relevant is the culinary habit of softening dry preserved bread in broth, brine etc. before consumption.  But it seems oddly specific that this very specific semantic development would have just accidentally happened to a verb with close similarity to kastua; and it is probably a good idea to analyze this similarity as having outright motivated the semantic development.
In any case, the conclusion still is that there is no morphophonological alternation a ~ o involved here: only two etymologically unrelated word groups, some of whose members have converged in meaning.
Since etymologists are usually mainly concerned with establishing connections between words, not in tearing them down, I would expect that there are people who fail to appreciate just how easy it is for words to show accidental or at least unetymological similarity, though. It is also often difficult or impossible to positively demonstrate that a given similarity definitely is accidental; and even producing calculations on the odds of accidental resemblance will be difficult, given how there are not really any “default hypotheses” about word origins that we could feed into these.
I believe there is however at least one method for demonstrating that accidents indeed happen: we can attempt seeking phonetically unlikely doublets, and see how easy it is to get these together, as compared with doublets that would seem to suggest some other, more phonetically expectable alternation.
Over some years, I have been collecting comparisons of this type from within modern Finnish, taking up cases of any imaginable phonetic variation (generally within the initial CV(C)C unit; anything going on in later syllables is often better considered “merely” morphology). Systematic surveying is difficult, and what I have so far is likely still biased in favor of alternations I have looked more into. Regardless the results so far are clear: even without giving too much slack for semantics, it is possible to get together at least a few surface doublets for approximately any alternation pair imaginable, while alternations with some actual historical motivation behind them generally generate larger amounts of doublets.
- a ~ e: lavea ‘wide’ ~ leveä ‘wide’
- ai ~ ie : taitaa ‘to know (a skill)’ ~ tietää ‘to know (information)’
- e ~ i: vehnä ‘wheat’ ~ vihne ‘awn’
- e ~ ää: retikka ‘radish’ ~ räätikkä ‘rutabaga’
- eu ~ uu: peuhata ‘to frolic, play rough’ ~ puuhata ‘to be busy, work on various small things’
- h ~ m: houkka ‘fool’ ~ moukka ‘boor’
- ha ~ e: harha ‘illusion, delusion’ ~ erhe ‘error’
- ht ~ v: kuihtua ‘to wilt’ ~ kuivua ‘to dry’
- i ~ ö: itikka ‘mosquito’ ~ ötökkä ‘bug’
- iu ~ ui: hiukka ‘little bit’ ~ huikka ‘sip’
- j ~ n: koje ‘machine’ ~ kone ‘machine’
- k ~ l: äklö ‘sickeningly sweet’ ~ ällö ‘icky’
- kk ~ pp: tukko ‘wad’ ~ tuppo ‘wad’
- l ~ s: lingota ‘to sling’ ~ singota ‘to shoot off’
- m ~ s: karmea ‘terrible’ ~ karsea ‘ghastly’
- m ~ ∅: muhkea ‘grand, bountiful’ ~ uhkea ‘voluptous’
- n ~ ∅: nilja ‘slime’ ~ iljanne ‘slippery ice’
- o ~ ä: vongata ‘to pester (esp. for sex)’ ~ vängätä ‘to pester (of children)’
- p ~ r: pöyhkeä ‘snooty’ ~ röyhkeä ‘arrogant’
- r ~ v: rako ‘cleft’ ~ vako ‘furrow’
- r ~ ∅: varsa ‘foal’ ~ vasa ‘calf’
- s ~ t: surma ‘death’ ~ turma ‘ruin, accident’
- s ~ ∅: kaista ‘stripe, lane’ ~ kaita ‘narrow’
- sk ~ v: rieska ‘flatbread’ ~ rievä ‘flatbread’
- t ~ v: tai ‘or’ ~ vai ‘or’
- uo ~ u: nuokkua ‘to nod off’ ~ nukkua ‘to sleep’
- ää ~ äy: ääri ‘edge’ ~ äyräs ‘brim’
This is a relatively representative sample, in that more than one of the above examples have demonstrably unrelated origins; more than one are also demonstrably related; several can be suspected to be the product of contamination in some direction; most however have no particular known explanation.
You can download the full list here (Unicode encoding; contents only in Finnish so far). In case you run into encoding woes, you can also try accessing this on pastebin. I have indicated a few etymological analyses so far, but most cases await fuller analysis. Further data can surely still be gathered as well. If anyone is interested in collaboration (analysis, adding in references to earlier literature, just adding in new potential doublets, etc.), feel free to get in touch with me.
For now I will not go into what kind of more detailed conclusions could be drawn from the data… though I imagine already simple eyeballing should be enough to highlight some features.
 On this topic, I often wonder how much of Latin we could in theory reconstruct from just the abundant loanwords it has left in modern western European languages. Or for that matter, if given no corroboration from the rest of Romance, would we be able to identify this reconstructed Latin as an early stage of French (rather than merely an extinct relative)?
 That the past tense of sortaa is typically sorti is then easily accountable as analogical, especially given that other verbs yet may show active vacillation, e.g. soutaa ‘to row’ : past tense souti ~ sousi.
 This example, for what it’s worth, is rewindable back to Pre-Finnic *kaja- ‘shine’ ~ *kajə ‘dawn’ with a slightly “weaker” alternation, and we could be dealing with some kind of an original derivation pattern in either direction; but this remains to be confirmed.
 I suppose an analysis as ‘to get so excited that you wet your pants’ might be theoretically possible, if we only knew of the modern sense of kostua.
 For further discussion of this word family’s history, see Hakulinen, Lauri (1940): Kostea ja kostua. Virittäjä 44.