I agree that it may be difficult to program a computer to find conjectures which it can't then prove, but can then be proven by humans. Much easier seems to be to find and prove theorems, or to search for counterexamples to conjectures.
That's why I am thinking at another possibility: explore mathematical constructions using computer programs, and from the observed patterns and regularities, suggest conjectures. So, not the program will be the one which suggest the conjecture, but he human user. Exploring mathematical constructions is done in experimental mathematics. On this Wikipedia page there are some examples of patterns observed when using numerical and graphical simulations, under the titles "Finding serendipitous numerical patterns" and "Visual investigations".