I don't see it as a priority. It would not be useful for me personally, but it's worth looking at it more broadly. Searching for
academia.stackexchange.com on Google Scholar, I find only 162 hits. This can be compared with the current number of questions, 41,937, not to mention however many answers there might be. The #hits/#questions ratio on Academia.SE is roughly an order of magnitude smaller than on MathOverflow.
Inspection of an ad-hoc selection of the search results for
academia.stackexchange.com reveals that
- some are just mentions of the site, e.g. to state where they collected data. These would not benefit from a "Cite" button. At least one of them is an ironic epigraph quote in a thesis, which might not need a full citation.
- some rely on posts instead of more authoritative sources, a practice we might not want to encourage further. An example is this paper, which cites this post but not the sources cited therein. (The paper also cites the post for various factoids not found within the post.)
- some are outright mistakes. For some reason, the question How do you cite a Github repository? has been cited at least twice instead of the intended repository. See this thesis and this IEEE magazine paper.
- out of the citations that seem valuable, a large part are in journals or fields where I would not expect TeX or BibTeX to be common. Would the bibtex format be familiar enough to those authors that they would know whether their reference manager supports importing such references? Or would they desire an expanded "Cite" button functionality?
In general, StackExchange don't want to add buttons on sites where they would never be used. On Academia.SE it seems likely a "Cite" button would be used at least sometimes, but given the overall low volume of existing citations and the debatable format of SE's BibTeX entries, I'm personally ambivalent about it.