I have the option to build a shrine in my basemant. My question is, which deity should I pay homage to? My character is a bosmer sneaky/stabby/snipey type. Does it matter much? I am thinking more of a RP reason rather than a practical one. However, are there practical reasons to have different shrines?

Does it have to be a deity recognized in Skyrim? As in, one of the Aedra or Daedra that already have shrines and/or cults in the province?

No real pratical reason to have them, just there if you want to venerate the gods, (the blessings you get are kinda useless), I'd say a bosmer out of the nine divines would worship Kynereth, godess associated with Kyne, who is probably the closest related deity to hunting, which links in to the sneaky bosmer thing