["To give up something" means something different from the meanings of "give up" in "give up trying" or " give up on trying"". I am making this a separate thread, but thank you for taking the time to try to find an old thread before you started an new one. ]
When you give up something to do something, you allow it to be used for a purpose.
"Thank you for giving up your time to help us."
You are thanking someone for using their time to help you instead of using their time on something for themselves.
Some might find the expression sycophantic, and it should therefore be used with care. It could be seen in BE as suggesting that the speaker is saying he is not worthy of the generous gesture of the person he is addressing.
In many circumstances 'Thank you for helping us' would be preferable.