Often "suggested the problem, and gave a few helpful comments" can be enough to merit a co-authorship. The student should probably ask his or herself: could I have written this paper without my supervisor's input?

When the student writes on a problem they have suggested to themselves, then the issue is more thorny. I think the standard etiquette is that the supervisor should be offered the option to be named as co-author (and as you rightly mentioned, it may increase the chances of publication, not least because the senior person knows the system better).