For a time, Tyson had a pet mouse called Rincewind: from the 23rd of April 2013 until 17th May 2014, when he passed away in his sleep.
He used to settle on Tyson’s left shoulder or on top of Lily’s head, who accepted that she wasn’t allowed to eat him. Bobby listened when Tyson told him to leave the mouse alone, but he would inquisitively nudge at the mouse to test if he could eat him this time. Bobby was also adamant that the rolling hamster ball was a moving snack and a toy for him to bat around.
Rincewind was a seal point fancy mouse with pink/red eyes. He was blind in his right eye and Tyson became far too attached for how short-lived he was. Rincewind was typically happy to be handled - though he could become overwhelmed - and, being very food motivated, was easy for Tyson to train through multiple tricks.
His most impressive trick, and one Tyson absolutely did not teach him, was to routinely escape his cage and ball. Even after Tyson replaced both with different types. Bar prompting Bobby’s prey instinct, Rincewind usually didn’t cause any trouble when he escapes. Either he went to pilfer the dog’s bowl or make a nest for himself in Tyson’s pocket.
Tyson is very fond of rats and mice, and to a lesser degree hamsters, but their relatively short life spans compared to cats & dogs, and his inability not to immediately become attached to anything with four legs, is why he typically doesn’t care for any.
If he didn’t have any dogs, or cats, Tyson would probably use the space he gained in his apartment to make a proper space to keep multiple rodents: probably the more social rats over mice. He might also choose fish or reptiles, but rodents are the most likely choice overall.