Mordell's reasoning might have been the following. If a curve of genus zero has a (smooth) point, then it has infinitely many. If a curve of genus one has two smooth points, it has a good chance of having infinitely many, as the difference will likely be of infinite order. Now, by Mazur's theorem, we know that seventeen points implies infinitely many for genus one. For higher genus, there is no such geometric reasons to get new points from old points, so a curve may have a million points and have no reason to have a million and one, so finiteness is reasonable.
I mentioned above in a comment a reason why most plane curves of degree at least four have no point (now I see that Scott has reproduced the argument in an answer, actually it may not be the same argument, but it is similar). I have no idea if Mordell had a similar heuristic, but it's entirely possible that he did as the arguments don't use anything he didn't know.
