Good question, and one that requires a different answer for just about every driver. I've met (and read) everything from the Lotus purists/fanboys, who answer every critique of the car with "it's fast in the corners!", to girls who bought one because "it's cute", to exotic owners with a stable of Loti to which it's just another car with its own set of characteristics and attitude.
Personally, I love Loti for the design ethos behind them.
1) Light, simple, minimalist.
2) If you want to drive this car, learn how.
3) Unique looks with honest heritage in the designs (e.g. the "smile" grill, the long history of the Esprit, etc. Ferrari lost this plot after the 355, and I've been peeved with them since)
4) Bang for the buck. Throw 20k at an Elise/Exige, and very, very few things will touch it in the corners or otherwise.
Specific to the Elise, I love it in addition to the normal "Lotus" reasons in that:
1) "Wear the car" design. It's hard to put to words, but I can't think of any most-weather road going car that offers better situational awareness as to where the car is and where it wants to go.
2) Targa roof. Targa roofs rock.
3) Extraordinarily unique chassis/construction method. Extruded pedals (and most everything else) are just plain cool.
4) Effortless to drive given the excellent feedback. I'm never left wondering what'll happen if I do this or that in a corner, the car tells me. VERY few others can say as much.
5) Dirt cheap given the design and capabilities.
The car has its issues, to be sure, and when I'm in engineer mode I don't gloss over them as some do. But when I drive it, if I'm completely honest, I really don't care. Few cars put as big a grin on my face as easily, and all of those that do cost a hell of a lot more and have just as many issues to deal with.