How do I give up my right as a mother of a teenage daughter?

Why would you want to do that? As a general rule courts will only allow someone to relinquish their rights to clear the way for an adoption or if the parent represents a danger to the child.

How old?
Does she work?
What do you hope to accomplish?

I don't think time will work this out and she is 16

If you can get her interested enough to get a job and move on her own you can get her emancipated.
A Teenager's Guide To Emancipation (http://www.larcc.org/pamphlets/children_family/teen_emancipation.htm)

Without a lot more detail, if she has proven herself a danager to you and others, there are some ways to give up your custody either to the father and;or to the state.

But don't feel bad, if you did not want to give up and walk away you would not be the normal parent of a 16 year old, I can remember my four order boys turning 15 and 16, it was hard.

Tried to inform her of those things but she rather run away & hang out with people I don't approve of