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There is an extended stay america off of bonney rd in virginia beach that is cheap if you don't want to stay on base. It is as close to the beach that you can get without paying an arm in a leg during the summer months!
BOQ at Damneck is literally on the beach. Just south of VA Beach right next to Oceana.
Not sure if they have a Navy Lodge there but I had a nice suite at the Q once. For a Navy Lodge, I would head to Little Creek. NAS Norfolk (Breezy Pt Q (Ely Hall)) is closer to Norfolk but a lot further drive to VA Beach.