I heard on Youtube (channel NowThis, video "Can Kosovo survive as an independent state?") that Russia prevents Kosovo from joining the UN because Russia backs Serbia, and Serbia doesn't want Kosovo to be independent.
So what's in it for Russia?
Chechnya and Russia's multi-ethnic confederated nature.
Russia hates the idea of independence movements and especially their success/recognition, given it is itself a confederation of ethnic areas some of which aren't all to happy to be in the confederation. The main parallel is Chechnya - which is also a rebellious Muslim-majority province trying to gain independence.
As such, recognition of Kosovo would create a precedence for such independence movements.
(yadda yadda Ukraine yadda yadda Georgia. Russia gets Gold Medal in hypocricy Olympics, which doesn't invalidate the reason above).
It's popular with Russians.
Russians are extremely xenophobic and partial to tribalism. That works on smaller Russian scale but also slightly larger Slavic scale. As such any support for brother Slavs plays well with russian populace; especially when Pindostan (Russian's insulting term for USA) is on the opposition's side.
Somewhat related to #2, historical entanglements.
Russia's backing of Serbia isn't exactly a new development - previously, a less-known conflict involving Serbia got Russia involved in certain unpleasantness with a couple of other countries on European continent.