"Weird" isn't the best way to look at it. Feeling weird when you find out she thinks of you outside of session, and feeling weird when she calls you by name in session makes me wonder if you struggle with attachment/emotional intimacy issues. I know I sure did. I think the absolute best thing you can do is tell you therapist.. sort of like: "I've had some reactions and I'm wondering if they are weird or something... like I feel awkward when you say my name, and when you told me you thought of me outside of session. I don't know what it means, but I wanted to tell you about it." Since she's working with you and knows about some of what's going on with you, she will probably help you explore why you have that reaction.

I don't know how long you've been in therapy, but that kind of conversation can really jump start the 'good stuff.'