So I've recently been asked if I could babysit a friends son who is about 10 month it would be 5 times a week from 5pm till 12 am

So how much money can I expect to earn she said I'll be getting paid through welfare ...can I ask to be paid a certain amount? Should I write a contract?

What country you live in?

Here welfare doesn't pay for 'baby sitting' unless you a certified business.

Library can help you learn about the process locally.

I live in California :0

Nothing, for welfare to pay you must be licensed and bonded

"Paid through welfare" sounds doubtful.