The name Tania is a girl's name meaning "belonging to the house of Tatius". Tania and discussed in our forums with posts like "Girls Makeover - Two Middles".
From the experts:
Tania is a Russian name that has been fully integrated into the American name pool, retaining just a touch of its Slavic flavor. It started as a pet form of Tatiana, which is beginning to be used in this country as well. Also spelled Tanya, as in country singer Tanya Tucker.