The Idiot was the only name given to Cat's father.
He was briefly depicted in the comic Red Dwarf Smegazine.
When the two factions of Felis sapiens built two arks and left Red Dwarf, they left behind those members of their society who were considered sick or disabled. They were abandoned to fend for themselves and slowly started to die off. The Idiot was one of those left behind ("Waiting for God").
He met and became friendly with a female Cat that was simply known as the Cripple, who was abandoned because she couldn't walk properly. They started to come together for companionship and eventually had a child who would grow up to be Cat.
The Idiot was a very religious Cat who would often remind his son of the Seven Cat Commandments (Infinity Welcomes Careful Drivers). Cat was surprised to learn that his father was an idiot (or "Jelly brain" as he put it), although it did explain why he had eaten his own feet ("Waiting for God").