I have looked for a list of pets that are available only:
1... at night
2... certain weather - eg sandstorm, raining etc
3... seasonal
I have:
Snowy Owl is only available during the winter months.
Qiraji Guardling only appears during the summer months.
Baby Ape only shows up when it is raining on that particular Island (separate weather from the rest of the zone).
Arctic Fox Kit Shows up during snow storms.
Silithid Hatchling Shows up during sand storms.
Restless Shadeling spawns at midnight, server time.
Shadow Sporebat only spawns at night (after 22:00 I believe) around Khadgar's tower in Talador..
Silithid Hatchling - found in Tanaris only during Sandstorms
Stone Armadillo - found only after dark in Desolace
Widow Spiderling - found only after dark in Duskwood
Please point me in the right direction

(Updated)