[The Only Real Estel]

I'm fairly certain that the hare started out as black. Legolas captured it in one of his many raiding party adventures, as recounted in The Complete Tale of Legolas & Co.'s Adventures; Vol. 3. Then, as I've found in my extensive search through the 1,000 page documentary Obsecure Side-Plots of The Lord of the Rings, he trained it to sniff out dwarves that happened to be traveling through the Mirkwood Forest. One particular dwarf had the singular misfortune of having most of his blue beard hacked off, so that Legolas could expierement with fur-transfusions. Thus, Legolas's hare came to be blue, though it can be argued that Legolas later found a way to make him purple, like other posters have mentioned here.

Pets in Middle-earth were rare, particularly rodents, which have a tendency to eat beatifully crafted furniture and nibble perfect flowing locks. One of the few characters who does possess a dog, at least in early draughts of LotR, is Aragorn, whose West Highland Terrier would earn him drinks by balancing peanuts on its nose.

Aragorn was one of the few to have a dog, but Sam was the only non-Haradrim being to ever own an Oliphaunt. His journey to the Grey Havens was partially brought on by the stress of trying to keep his 'Ollie' (as he named it), & keeping all of his beloved trees safe, no easy task. In The Supposed Tales That Are Still Unfinished, it says that Sam brought Ollie across the water with him on a barge, & so the two came to the Undying Lands together. That, of course, is only partially verified-take it with a grain of salt.