I wouldn't say that it "forms the entire basis of Special Relativity." It's just one way of introducing the topic. It has the advantage of being relatively nonmathematical.
Interesting question. I suspect it was invented about 70 years after Lorentz's work. The first place I encountered this presentation was in Hewitt's textbook Conceptual Physics, which dates back to 1987. This paper http://arxiv.org/abs/0705.0941
says, "The light clock is a pedagogical device used by many authors for deriving the formula that accounts for the time dilation relativistic effect," and gives two references, the earlier of which is to Space and Time in Special Relativity by Mermin, dating back to 1968. Amazon let me peek at the relevant part of the book with their "look inside" feature. (Mermin's more recent approach to the pedagogy of SR is given in his newer book It's About Time.) This paper http://arxiv.org/abs/physics/0505134
points to an earlier use of the idea, in 1963 in the Feynman Lectures (section 15-4).
The problem with the light clock as an introduction to SR is that it requires Einstein's 1905 axiomatization of relativity, which, with the benefit of 107 years' hindsight, inappropriately singles out light as having a special role. We have a FAQ about the different possible axiomatizations: http://physicsforums.com/showthread.php?t=534862
A more nitpicky objection to the light clock is that I think most presentations fail to justify a hidden assumption, which is that the length of the light clock is the same in both frames. This assumption would be incorrect if the light clock were oriented longitudinally rather than transeversely.
There is clearly a close affinity between the Michelson-Morley experiment and the light clock. Feynman pretty much develops it this way (and also doesn't cheat on the issue of longitudinal and transverse length contraction).