The Siamese cat is an occasional drop from Cookie in the Deadmines. It dropped when we went in, but my brother got it.
You can run the Deadmines starting at level 17, but I strongly suggest 18-24. If you have a pick-up-group, you should be 22-24, to minimize dying.
Cookie is on the ship at the end of the instance, after you start fighting pirates and just after you board the ship (after fighting a Tauren boss). He's on the northern tip, behind some crates. If you look on my image's minimap, I'm standing close to where he starts -- if I were to walk forward and around the crates ahead of me, I'd see the spot.
The cat does not always drop, but it drops often enough that you can usually find it on the AH.