Jason was one of the Resident's pet dogs.
When the Tourist, Miranda, Shona Daniels, Pathway and Ashlyn Oswin visited the Resident in one of the houses she owned inside the “finished” 10,000 Dawns, she had two pet dogs with her: Jason and a Golden Retriever. Both dogs immediately took a liking to Shona, licking her face profusely. (PROSE: The Leader of the Band)
Behind the scenes
The Resident having two dogs, one called Jason and another one only ever called “the Golden Retriever”, may be a pun of sorts on the legend of Jason and the Golden Fleece, also alluded to in the title of the Lady Aesculapius episode Jason and the Astronauts.