1. Versatility. They are eminently suited to almost every style: hard rock, punk, pop and country of course.
2. Simplicity. Two pickups, two knobs, three positions. A Tele is built for utility with no frills added.
3. Learning. I've never made so much progress in my playing as I have when practicing with a Telecaster (not counting acoustics). It is an unforgiving instrument that demands precision.
4. Sensitivity. I've never played a guitar that is so sensitive to control adjustments or playing technique. Roll off just a touch of volume or tone and the sound changes dramatically. Different picking techniques yield radically different tone.
5. Coolness. There's something inherently cool in a Telecaster. It just looks...perfect.
I also own Les Pauls, Stratocasters, and a Rickenbacker, and I'd never want to part with them. There are particular applications where each of those guitars provides something unique and desirable. But if I could only keep one, it would have to be my Telecaster. Even if it isn't better in the areas where the other guitars really shine, it can still do those things in close approximation. The reverse is almost never true.