Virginia is an awesome choice! Gigi? Vivi? Gina (like Geena)?
Ginny is CUUUTE though. That's what I would do! Also, I have a daughter with a name that lends itself to one nickname out of a few options (I.e, Margaret to Maggie vs. Margo or Greta). If you want the less common nickname, be FIRM and establish the nickname immediately. Otherwise you're fighting an uphill battle.
http://nameberry.com/nametalk/threads/73109-Nicknames-for-Virginia