One of the bigger open problems of Finnic historical phonology is the shift *o > õ in Southern Finnic.
The non-front non-open illabial vowel õ found across Southern Finnic — the exact realization varies from /ɤ/ to /ɨ/ — most regularly corresponds to Northern Finnic e in words of back harmony; e.g. Estonian mõla ‘paddle’, põld ‘field’ ~ Finnish mela, pelto. If these cases should be reconstructed with front *e or back *ë in Proto-Finnic remains disputed, but the correspondence pattern is fairly unambiguous. 
Frequently, though, õ is also found in correspondence to NF o, e.g. in Es. õlg ‘shoulder’, õlg ‘straw’, hõbe ‘silver’, kõrv ‘ear’, lõhi ‘salmon’, sõrm ‘finger’ ~ Fi. olka, olki, hopea, korva, lohi, sormi. Cognates from elsewhere in Uralic, or in loangiving languages, fairly consistently indicate that Northern Finnic retains here the original state of affairs, and anything along the lines of a Proto-Finnic central rounded vowel **ȯ probably should not be reconstructed.  However, o also frequently remains; e.g. Es. oja ‘brook’, ots ‘forehead’, kolm ‘three’, tohtima ‘to dare’ ~ Fi. oja, otsa, kolme, tohtia.
The task, then, is finding the conditioning for the delabialization *o > õ. It’s been observed already long ago that no easy solutions are available. Which words show this change and which do not varies greatly already depending on the language variety in question. Previous analyses by e.g. Viitso and Raun have distinguished some 4-5 main distribution patterns. It seems probable that the issue cannot be fully answered without detailed analysis of the dialectal diversification of Estonian. Interdialectal loans, especially to and from the literary standards,  have probably also muddied the original distribution quite a bit.
This all regardless does not prevent partial progress being made. In particular, Livonian is known to be a clear-cut case: delabialization is here found effectively solely in the diphthong *ou and in the sequence *ovV, which entirely regularly yield õu / õvV.
This in mind, a look at how the change has played out in its other geographic extremum — Votic — might also be fruitful. This does not seem to have been done, though. Lauri Kettunen’s Vatjan kielen äännehistoria (1915) does not even attempt an analysis, and in case anyone else has since him examined the historical phonology of vowels in Votic in similar detail, I am not aware of it.
I will not be presenting a full analysis here, either, only a couple of hypotheses, based on a relatively quick look-over of the lexicon of the Mahu dialect. A look into data from other, fuller-documented dialects of Votic (these days well-summarized in Vadja keele sõnaraamat) will be necessary to confirm or deny the following ideas.
The main impression that emerges when examining Votic on its own is that attempting to determine conditions for *o > õ is probably the wrong approach. Delabialization occurs much more often than not, in almost any phonetic environment imaginable. Only two sources of modern o are immediately clear: firstly, long oo is entirely unaffected by delabialization (as also elsewhere in Southern Finnic); and secondly, recent loanwords, from Ingrian, Finnish, and Russian, consistently retain their short o. There appear to be a number of other word shapes where short stressed o is probably inherited — but they are quite few, and what seems like an approach worth exploring is that this development has been conditional, while delabialization would be simply the default reflexation.
One particularly promising environment are words with original *o also in the 2nd syllable. Clear cases are the words koto ‘home’, roho ‘grass’. Loaning is not an option for either of these: the former was in Eastern Finnic (and standard Finnish) replaced by the form koti, while the latter shows vowel shortening before h, a sound change absent from Ingrian and Finnish.  The same rule could be used to account for some other words as well: kokottaa ‘to bawk’, mokoma ‘such’, orko ‘valley’.
Two very interesting words moreover turn up: opënë ‘horse’, toro ‘acorn’. These have not had original *o at all, and they rather derive from Proto-Finnic *hëpoinën, *tërho (or *tëroh?) — cf. Finnish hevonen, terho. This suggests that the history here was actually not the retention of stressed *o before a subsequent *o, but instead a later back-assimilation *ë-o > *o-o (and in ‘horse’, this was presumably folloed by an even later assimilation *-o-ë > /-ë-ë/). 
This opens also a new possibility for etymologizing koko ‘pile’. Given Finnish keko, perhaps this similarly derives from earlier *këko; in which case the synonymous Fi. koko, and its other Finnic cognates, would seem to turn out to be loanwords from Votic? (If the same would go also for koko ‘whole’, koota ‘to gather’ is not clear. I could also imagine these being derived from *kokë- ‘to check, e.g. traps’, perhaps via a meaning ‘to collect’.)
But on the contrary: no assimilation of this sort is seen in e.g. nõvvoa ‘to advice’, põlto ‘field’, sõtkoag ‘to mix dough’, võrkko ‘net’, võso ‘sprout, young tree’ (all five inherited words of the *ë-o type). The first could be maybe explained as being due to the general ban on ˣ/-ouC-/, and the last two similarly due to the ban on ˣ/voC-/. But ‘field’ and ‘to mix’ resist easy explanations. I wonder if both of them being loosely agricultural vocabulary has any relevance.
Another environment where o could perhaps be regular is the context C_(C)kka. In my sample, no examples with õ are found in this position, but instead there’s a full handful of examples of o: hoikka ‘thin’, kokka ‘hook’, kolkka ‘corner’, nokka ‘beak’, rokka ‘cabbage soup’. At least the first is evidently an Ingrian loan, on account of retained /h/, but given the average proportion of such loans, I am not sure if we should expect the same to be the case for all of these.
It’s unclear what phonetic motivation could exist here, though, since a lone coda /k/ is not enough to block delabialization; e.g. *oksa > õhsa ‘branch’.
A third case where o fairly often remains is the diphthong *oi. From Mahu there are five positive inherited examples: koira ‘dog’, koivu ‘birch’, moisio ‘manor’, poika ‘son’, poiz ‘away’; and three negatives: õikõa ‘right’, nõissag ‘to rise’, sõittaag ‘to scold’. The sixth could suggest a shift *oi-ë > õi-õ; and the seventh may have something to do with how the rest of Finnic rather indicates earlier *ou (Es. tõusma, Fi. nousta), perhaps suggesting something like *novise-?  But no especially clear pattern seems to emerge here.
The issue of õ will for now still remain unsolved, but at least it is clear that some clues will emerge as long as one is willing to look for them.
 Given that this correspondence mostly occurs in loanwords from Indo-European that once had *e-a, *e-o or the like (e.g. pelto derives from Proto-Germanic *felθō), it’s clear that there has been a retraction from earlier *e at some point, but what is not clear is if we should assume this to have been an areal Southern Finnic innovation, or a common Proto-Finnic innovation followed by a backshift in Northern Finnic. The question is out of the scope of this post, but suffice to say that I narrowly lean in favor of the latter (mainly per some indirect arguments from relative chronology, which however rely on a few so far unreleased ideas of mine).
 In principle I am however open to the possibility that in some cases Southern Finnic might regardless retain the more original state of affairs, especially since by now we know that Proto-Uralic also featured a non-front non-open illabial vowel *ë. This is normally continued by Finnic *a, but it’s imaginable that some conditional exception developments exist. One particularly interesting gap to observe is that while PU seems to have allowed a fairly wide variety of vowel + semivowel combinations, no cases of the sequences *-ëw-, *-ëj- have been reconstruted so far.
 That is, both the modern North Estonian-centric standard, mainly based on the Tallinn dialect, and the old South Estonian-centric standard, mainly based on the Tartu dialect.
 Somewhat mysteriously this feature is however found also in Veps. — In principle it’s of course also possible that shortening before h is simply later than *o > õ.
 Interestingly enough, also Standard North Estonian seems to have this change in hobune ‘horse’. Yet ‘acorn’ remains tõru. Could early loaning from Votic to Estonian be involved?
 This reconstruction would, then, rather look like a derivative *nov-ise-, and we could ask if the root is somehow connected with PU *ńoxə- ‘to chase, follow’ (whence also PF *nou-ta- ‘to fetch’). But the semantics do not seem to work even elementarily, especially since *-ise- is well-attested only as deriving onomatopoetic verbs of all things.