"With whom" is grammatically correct, a little formal but that is acceptable. You could improve your phrasing with some of the social language:
My friend: I was there together with (Jon). I wanted to speak with him.
Me: Sorry, I didn't catch the name. With whom did you want to speak?
I've added an apology (I'm British, we say "sorry" a lot) and I've cued the question with an explanation ("I did hear", not "I heard but I don't believe it"). It would be equally correct to ask "Who did you want to speak with?" A full sentence here is more friendly.
