T-shirts get worn in a variety of different situations, which demand radically different materials.
About fifteen years ago I discovered microfiber, which totally transformed my athletic endeavors. No longer would my workouts (previously performed wearing regular cotton t-shirts) end up with a soggy rag draped around my torso. Microfiber does a fantastic job of "wicking" sweat sway from the skin, allowing it to evaporate quickly, cooling my body and allowing me to a) be comfortable and b) perform at my best. Short answer: If you exercising in anything other than microfiber - you are making a huge mistake.
In a "casually elegant" situation, you would be hard-pressed to come up with something more appropriate than a well-cut, well-fitting silk t-shirt. Paired with a deconstructed jacket and fashionable jeans, this is a look that will permit entry to the swankiest restaurants and clubs, but without making you look like a stuffed shirt.
For the rest of the time? Cotton. Painting the garage, building a DVD-shelf, or scrubbing the bathroom are activities best undertaken in an old cotton t-shirt. For hanging out with friends or casual dates - a nicer (no sweat-stains or ragged collars) cotton t-shirt under a button-down shirt looks fantastic. And the cotton t-shirt serves well as an impromptu night-shirt, should circumstances require it.
Summary: There is no ONE "best" t-shirt material. It all depends on what, and where, you're doing it.