I am an adoptive mama of two different children from two different first mothers. We have different degrees of openness with each of the first moms, more due to their choices than ours.
DS went home with first mom L for 2 1/2 months before she made the decision to place him. She of course gave him a name and he was called that until placement. DH and I chose to change his name. I really wanted to give him as his middle name the first name that L had chosen for him, but DH adamantly refused. I regret that I didn't fight DH more on this. L and her parents and siblings insist on calling DS by the name that they gave him and knew him by during his time with them. This is very confusing to him as he is too young to understand adoption. It hasn't been a huge issue because we have only seen them a few times.
With DD, her first mom D let us name her from the get-go. Even though I encouraged D to choose her own name, I have to confess that at the time, I was grateful that she declined. Not for the good of DD who I think would probably appreciate in the future having been given a name by her first mom. But for my own good as I wouldn't have to deal in the future with the issue of her wanting to be called a name other than the one that we gave her. DD has my name as her middle name, so no name connection to her first mom at all. :(
I wonder how DS will feel about the name thing later. What if he prefers the name that L gave him? I guess we will cross that bridge when/if we come to it. As I already said, I regret that we didn't keep L's name for him as his middle name. We removed any name connection to his first family and that just isn't right.
I have to add that I don't feel that any of us--especially the first moms--had good counseling on this. Mostly because we used a private attorney and, honestly, I don't think that any of it was handled in the most ethical fashion. I know so much more now about open adoption than I did then.