You can change it via the Profile option in the drop down menu available from your avatar in the top right corner of your HipChat desktop client (web or native client). Here is a quick video that shows the process.
Your @mention handle is your username. When accounts are created the local part of your email address is used as the default username. So if your email address is JohnDoe@example.com, your username would default to JohnDoe, and therefore your @mention handle as well.
I don't know how this works in HipChat, but in Confluence only a site admin can change usernames. It's a trivial task, but you do need permission to access User Management, which only site admins can normally do.
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