First, it has to hold water, then it has to fit to the tea, and finally, it should please my eyes/fingers/lips.
When I'm out and about, the lids/cups of my thermoses make a serviceable sturdy cup, but their lack of aesthetics actually detracts a bit from the experience.
Fit to the tea means size and shape and thickness: size, not too big or too small for the typical volume of each infusion--a little larger is ok for a few larger infusions of sencha or white tea, but for a dozens of infusions of a very nice puerh, I prefer a very small cup. Especially for the young shengs that can get more bitter as they sit and cool, I prefer a shallower wider vessel so it cools quick enough to toss down the infusion. Thicker for hot-brewed teas is a must, or else a shallow shape to help dissipate the heat. But cooler brewed teas can go in thicker or thinner cups.
Shape, feel, appearance & color--have to be aesthetically pleasing. The lip of the cup needs to have areas that feel soft to my lips. The shape should feel comfortable in my hand: I've found a few cups that pleased my eye were just too big in the wrong places to feel solidly in my grasp one handed, and those do not get used much. I've got cups I like very much that are slick and mass-produced, but having an artisan's story behind it also can add something to the experience. And so far the aesthetically pleasing features include short, tall, curvy, angular, thickly glazed, thinly glazed, matte, iridescent, earthy, brilliant, sparkling, and subdued. Much as a glaze adds to the experience, I'm quite sure that one of these days an unglazed bizen-yaki will take subtlety to a new level.
Now drinking Aoi sencha from a very small, short, cylindrical cup with crawling shino glaze, unglazed patches, deep russet interior, and the very small cup also functions as an excuse to bring out the lovely yuzamushi pitcher to hold the excess volume of tea.