Lovasz's proof of cancellation in certain classes of finite structures still bewilders me; I can only imagine that he found the proof first and then came up with the theorem afterwards. The basic idea is to look at homomorphisms between a given structure and a sequence of other structures. A comparison of two such sequences involving structures of the form AxC and BxC can be taken to a comparison between A and B. The condition that there exists a one-element substructure is used to show a certain nontriviality of the comparison, and a few more details result in showing A is isomorphic to B if(f) AxC is isomorphic to BxC.
I should have asked Lovasz how he came up with the proof; I am confident that most people would not be able to come close to the method independently if they were only given the theorem statement. (Not to mention the analogous statement of unique nth roots in the same class.)
Gerhard "Ask Me About System Design" Paseman, 2010.12.09