It's from the second episode of the tenth season of Friends. Here is the scene. It's at 1 minute and 49 seconds.
Chandler: I'd love to, but I gotta get back to talking to your
parents. They're telling us all about how they adopted you.
Owen: What?!?
Chandler: What?
Owen: I'm adopted?
(Chandler tries to come up with a good answer.)
Chandler: I got nothing.
Best Answer
In this context, "I got nothing" means "I don't have anything to say". It could mean that you choose not to add anything to a discussion, but often in this kind of context, it means that you have tried to think of something to add but were unable to.
As you point out, this is his response after he tries to come up with a good answer.