Miss Tuffsy was the previous snail owned by Granny before she took in Gary. She is only mentioned in the episode "Have You Seen This Snail?" It is implied that Miss Tuffsy died due to Granny overfeeding her to fatten her up to be eaten. It is also possible that Miss Tuffsy discovered Granny was trying to fatten her up to be eaten so she ran from Granny.
Miss Tuffsy herself has never been seen in the show. However, it is possible that one of the snails in the picture on Grandma's wall could be her.
Role in episode
Granny mistakes Gary for Miss Tuffsy and treats him like if he were her: they go shopping, go to a beauty salon, listen to music, and watch movies. After Gary realizes her plan, he escapes her house and goes back to SpongeBob. Granny follows him and then picks up one of the Alley Snails that Gary met in the alleyway, thinking again they are Miss Tuffsy.