One problem that perhaps fits in this category is the exact solution of the hard hexagon model by Rodney Baxter. In this case computer calculations revealed surprising patterns which suggested that the model was solvable, but it required someone like Rodney Baxter to first recognise these patterns, and then go ahead and find the solution. A large grey area in the nature of the question is the extent to which the computer "discovered" the solution.
This discovery is described in Ch. 14 of Rodney Baxter's book "Exactly Solved Models in Statistical Mechanics".
There must be other examples of computer discoveries in the field of solvable models in statistical mechanics, as to some extent this is the nature of field. However, to me this example stands out, as to the best of my knowledge there was no a priori expectation that the model was solvable before the computer calculations were performed.
