I actually can't think of any words to describe what Regina is thinking but I'd imagine she's feeling possessive, a little jealous, thêm than a little neglected(? not quite the word I want but if I think of it later I'll fix it. xP).
I see a lot of irritation and frustration against Emma for her claim on Henry at this moment but honestly I can't blame her. Regina may tình yêu Henry, and I believe she does, and both she and Henry regard them as being mother and son but that's not the whole situation, especially from Emma's point of view.
One of the first things Henry says to Emma is that Regina doesn't tình yêu Henry and that she only pretends to. Later on in the Pilot Emma asks Regina whether she loves Henry. Regina says "Of course I tình yêu him." but something in that answer encouraged Emma to stay instead of going back to Boston. Throughout the first season Regina mistreated Henry and used him in her vendetta against Emma. Eventually her spite towards Emma led to Henry almost dying. And that's all just season 1, I don't blame Emma for not thinking Regina is a suitable guardian for Henry because she just isn't. She can be, certainly, just not until after a lot of therapy and acts of redemption to prove she's changed and has put her toxic past with Henry's grandmother behind her.
@Minervadawn, sorry I didn't mean to rant - I know you've heard all this before.