Count Snoozy (per captions for Drac's initial mention, also spelled Count Snoozey when Mavis talks to herself later) is a vampire count who Drac tells Mavis about in Portrait of Mavis as a Young Vampire. He fell asleep while having a portrait done, which messed up his street cred.
Drac does not explicitly say whether Snoozy was a Dracula or if his portrait was in the Hall of Draculas
It is shown he fell asleep clutching a teddy bear, which was painted.
After Mavis messes up her own painting (showing her drooling) she imagines him laughing at her briefly along with others. She says "I can't becoming the laughingstock of the family" which seems to imply she might consider Snoozy to be family based on Drac's story. In which case his name might be Snoozy Dracula.