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I've lived in the Bay Area my whole life and I'm also getting my bachelor's from a Bay Area university. I'm aiming for biglaw and would prefer to work in California however I would have no problem working elsewhere.

Nothing wrong with shooting for Bay area big law and if you miss taking the NYC job and trying to lateral later. If you get the grades you'll have a chance but nothing is guaranteed because it's a popular destination and there just aren't very many jobs. If you can get into Berkeley (and certainly Stanford) that would probably be the better choice but obviously that's not a choice everyone can make.

The Bay Area market is quite tight. Your ties help, but keep in mind that the class sizes are generally much smaller than other markets. I know some classmates of mine at Berkeley who wanted to stay in the Bay Area but ended up taking offers in NYC/LA just because there are more offers to go around coming out of those markets. But since you're willing to work elsewhere, just make sure to apply in multiple markets when the time comes and don't put all your eggs in the Bay Area basket and you'll be fine.