Well, I've tried the "Can I help you?" with absolutely zero effect on her turning around and staring. I guess my next step is to ask her to please stop and to turn around when she's doing it.
Now, she is also doing a new thing. She likes to tell me when I have a missed call. Our phones are set up so that if someone is receiving a call, a little notice scrolls across the bottom of everyone else's phones. It's an unnecessary thing that was left over from when they only had 6 employees total here, but IT has never re-configured the phones to not do it. So every time I come back to my desk, if Nancy sees that my phone was ringing, she says "Sara tried to call you. You also had a call come in from someone named Tonya." If I'm at my desk and my phone starts ringing, she will say "Oh, it looks like Maggie is calling you. What does she want, do you think?"
Yesterday I was actually on the phone and someone tried to beep through, but I couldn't switch over to them because the call I was taking was important. While I was still on the phone, Nancy loudly said "Hey, it looks like Wendy Smith is calling you!" and when I hung up, she said "Wendy Smith tried to call!" Yes, I know..I can see when a call is coming in on my line just like I can see a list of any missed calls I had when I look at my phone. Our system is also set up to send an e-mail notification of any missed calls.
I told her "Nancy, I can see when I have a missed call so I don't need you to inform me when someone has tried to call me". But she continues to do it. I'm glad she hasn't yet figured out that she can pick up calls from my line.
They are discussing re-arranging offices at work, so I put in a discreet request to the HR contact for our campus that if they do indeed re-arrange offices and departments to please swap Nancy with Mary, who is another new assistant who is much quieter and far less into what everyone else is doing.