When my mother saw

her first grey hair she

thought she'd dye. Explain the pun

It's referring to dye as in color dye.

Explanation:

She saw her first grey hair, which is not her natural color I'm assuming. So when she said she thought "she'd dye" the pun is referring to "dye" as in, "dye your hair". The phrase has two meanings to it, she thinks she will die, literally, but also there's a pun because her hair changed color so "dye" as in a color dye.