Alex was actually arguing with her. He had a real opinion on something.
Claire was completely amazed. How could she say no when he wanted something and had come up with a semi rational argument for it?
But this was so weird! A man that she wasn't related to at all aside from a mutual agreement wanted to take on her last name! Once she had repaid him for his protection by helping him learn how to function in the real world he would leave but he would always be stuck with the name 'Conway' if he did this.
He already planned to get different identity documents when this was all over because he liked 'Alex' better than 'John' and didn't want the Agency to be able to track him. If those documents said Alexander Conway, that was who he would be for the rest of his life. That was a pretty big decision to make on a whim.