Sardonic, a little cynical, perpertually unhappy and eternally grateful to his friends who try to cheer him up - what's there not to love about Eeyore?
While the casual observer of Eeyore would see him very depressed, he actually has a heart of gold. Capable of extraordinary compassion, Eeyore can make plants grow by showing them a little love where others (Rabbit, the professional gardener) failed.
The Milne version of the character never has him happy, but Disney managed to subvert this gloomy donkey. The FantasyLand version has him actually smiling from time to time, especially on the stuffed animals sold at that happy place (to quote Walt).
So, bottom line, what I like about Eeyore: to most - he is what he seems to be, but those that know him best appreciate the colorful nuances of his personality, and love him deeply for them.