Why would I need a copy of my mother's birth certificate? she says to keep a copy "just in case" under what circumstances would it be useful?

I'm guessing, but anything where you need to work on her behalf and establish her identity when she is no longer able to do so might need that (You probably also need power of attorney). Things like establishing or retrieving new government benefits, accessing bank accts, cashing in foreign bonds (I know Israeli bonds request that), stuff like that.

her death or incapacitation. If she's able to explain that to you, she should. Ask her what's she's thinking about.

Applying for a passport, you need all of her details.

Many circumstances.