A small Eastern Finnic word family with no known origin is Southeastern Finnish nauka, Veps ńoug ~ nåug ‘fish slime’. Possibly Karelian ńaugu ~ ńauguheinä ‘bur-reed’ also belongs here. This seems acceptable to me: the usage example in KKS comments that lye from the ashes of the reed has been used for washing hair, and lye solutions are typically some degree of slippery or slimy. The latter compound ‘ńaugu-hay’ is also telling: the independent word may have been first lost, then extracted from this compound as referring to the plant itself and not its use. Regardless of if we accept in the Karelian word though, a common proto-form *nauka could be reconstructed.
But I wonder if these words actually rather continue Proto-Finnic *nolki ‘slime’ (> Estonian nõlg, Livonian noļg, Karelian ńolgi, Ludian ńolg). No Veps reflex appears to have been recognized. The predicted Veps reflex, however, would be indeed exactly n/ńoug. This suggests to me that there’s actually only one word family in here, and the Finnish/Karelian words suggesting *nauka are relatively late loans from Veps. SSA notes that the Veps word has also been loaned to local dialects of Russian (нявга ‘fish scales’).
Why then -a- in Fi./Krl. and not -o-? I suspect one solution could be etymological nativization gone awry across the Veps / Karelian dialect boundary. Northern Veps is characterized by the raising of inherited diphthongs: *au > ou, *ou > uu. A word that got its ou instead from *ol would thus end up recognized as rather continuing underlying *nauka than *nouka.
This kind of an explanation could also be how the declension type of the word got modified to an a-stem in Finnish. The Veps and Karelian words appear to continue a derived stem *nolk-u; — and in Livvi, at the southwestern end of the Karelian dialect continuum, word-final *-a is normally raised to -u. If the word was at one point known all across the way from the Gulf of Finland to Lake Onega, then during its diffusion southwestward, someone may have again etymologically hypercorrected a Livvi *naugu into Finnish nauka. (But other possibilities exist as well, such as back-derivation, or loaning from a not fully apocopated form *ńougə in Old Veps.)
Both *nolki and “*nauka” are also among the words where Eastern Finnish appears to retain PU *ń-. More likely this is however not of especial etymological signifigance, but merely later expressive palatalization, given the regular loss of palatalization in more widespread words such as PU *ńëlə > PF *nooli > Krl. nuoli ‘arrow’, PU *ńalə- > PF *noolë- > Krl. nuolla, Ve. nouda ‘to lick’, PU *ńoxə- > PF *nou-ta- > Krl. noutoa ‘to follow’.