"Do my best for something to happen" is grammatically correct?
“I did my best for them to add (something) in the list” is that a grammatically correct way to convey the intended meaning, which is (I convinced them) to add that thing to the list?
Your phrase would be easily understood but this is not common usage. I suggest:
I did my best to convince them to add (something) to the list.
Other possibilities:
I did my best to persuade them to ... I did my best to get them to ...