A g-stare occurs when someone requests to add you to their g-mail g-chat list. You accept, expecting a "hello" or "how's it going," but then they don't say anything. However, you can see their handle is online and active. The equivalent of someone you know saying "hi" on the street, and then staring blankly at you until you walk away.
Gerry added me to his gmail g-chat list, but instead of starting a chat he just sat there idle. He must be trying to build his chat list, or he was just up for a good g-stare.