Cats have a long and illustrious history on ships. Sailors have been bringing cats on board for at least 9500 years. Personally, I don't see why that tradition would change 1286 years in the future. Besides, cats would probably serve the same invaluable role as they did on the ships of yore, which is keeping the rodent and pest population down. Only instead of eating the foodstuffs you were trading 3,000 years ago, the pests would be nibbling on wiring, potentially causing issues with your ship in deep space, which, as we all know, would be a huge problem. Furthermore, rats and other pests tend to carry diseases which could be spread from port to port. Cats would also cut down on the risk of that.
Now that I've got the RP reasons to have cats in Elite out of the way, let's talk about implementation. It probably wouldn't be that hard to make an animation of a cat that does simple things like lay across the top of your dash, rubs up against your character's legs, and walks around the cockpit of the ship. For that matter, you probably wouldn't even be able to see the cat doing most of those things if you weren't in free look mode. However, I think that this would be a very cool idea, and it's something I would gladly spend ARX or perhaps even real money on.
The idea was brought on by my cat laying herself across my keyboard while I was trying to get ready for a session. So after a minor annoyance (and a half hour spent petting and cuddling my cat, because she is awesome and deserves it) I had an idea that I was excited to share with the community. I hope you all think this would be as cool as I do.