He is not premed, he is math major, going for Ph.D. in mathematics
If he's interested in getting into a good grad school then it is more important that he kicks the math subject GRE's ass. If he wants to be competitive then he needs to score 80%+ (and he should score perfectly/near perfectly on the math portion of the general GRE). What I've heard is that less than 80% raises red flags at the top schools. A good GRE is a prerequisite--the LORs are most important.
As for undergrad research, I've heard that it is a waste of time. It doesn't really give you a sense of what doing mathematics professionally is like. It'd be better to spend that time going over all the past coursework in order to do well on the GRE and to show his mathematical maturity and curiosity in class so he can get good LORs. You might argue that he could get good LORs from research but he can get better ones from his classes. If he insists on undergraduate math research, then look up 'math reu' on google to get a good sense of the opportunities. He can also study abroad in France or Russia--they tend to have some good programs and it'd diversify his outlook.
He should also know that he's playing with fire going into math professionally. To paraphrase Paul Halmos, if you have any doubt about going into mathematics then you shouldn't.
The career options are research academia, teaching, and industry (typically finance). Research academia is hard and he should be prepared to be disappointed for the rest of his life because there will always be people much, much smarter than he is. Maybe he's brilliant and he'll do great but he needs to be ready to see that he may not be up to par. I can't tell you how many times I've heard tenured profs bitch about how hard they worked and they didn't get a job at Princeton, Harvard, Berkeley, etc. But if you're good then academia is probably the best job you could ever imagine.
Teaching is a good, stable job--it pays, well, enough to survive. It isn't too hard and it is quite fulfilling--though there is a limit to how much emotional satisfaction you can get from teaching calculus or, if you're lucky, upper-division classes like real analysis or number theory.
Industry pays really well (many Wall St. firms, before the recession, would give $200k starting salaries before bonus--which could double your salary). These jobs are competitive and the interviews can be a bear--e.g. he'd have to answer questions like 'if you're a surgeon and you have to operate on three people but you only have two pairs of gloves, what do you do?' and you have to think quickly on your feet to come up with a good answer. The impression I get is that most people work in industry for 10-20 years then retire because the job sucks.
