I recently came across a friend who is applying for a PhD in my lab. He happens to be from Brazil, on an F-1 visa. He was told by my advisor that his citizenship status affects the sources of funding available to support his work. I was always under an impression that the principal investigator applies for research grants and funding does not depend on visa status (if any) of the students under his guidance. Was I wrong in making this assumption?
It absolutely depends on the source of funding. Each organization has their own rules. By and large, the NSF (since you mentioned F-1 visas I assume this is US-centric) allows foreign students to receive funding from grants, but this intent has to be disclosed at the time of submitting the grant request.
DARPA has slightly tighter rules although I'm reasonably sure foreign nationals can be funded. But if you're from a "country of interest" like Iran, there can be other complications.
Finally, while these are not PI-requested grants, the NSF-Hertz graduate fellowships are exclusively for US citizens.
p.s This even applies to PIs. There's a large DARPA program that you're not eligible to apply for unless you have a SECRET clearance, for which a necessary but not sufficient condition is that you're a US citizen.
The question (or the supervisor's answer) does not indicate clearly if the supervisor is planning to incorporate your friend into one of his grants as support personnel, or is planning to have your friend write his own doctoral fellowship grant.
For the first option, the student's visa status does not matter in most of the cases. I have seen CDC and NIH grants used to fund foreign students. My PhD was also funded by NIH (I was not listed as an investigator) when I was holding an F1 visa.
The only wrinkle I can think of is that, the PI (principal investigator) might have budgeted a full time salary of, say, $50000 a year for an assistant. Due to F1 visa's limitation, the student has to be a full-time student, which means he cannot get full-time salary as a worker simultaneously. The usual approach is to resubmit a minor adjustment and change the budget into $25000 part-time salary and $25000 scholarly support. At this point, some problems can occur as some grants might have specified that they wouldn't fund school fee (which means your friend may have to look for scholarship to supplement his study. But he will have some stipend from the grant), and the agency may also question the feasibility of cutting a position's time in half. etc. Otherwise, I cannot think of other significant reasons.
For the latter case, then visa status matters a whole lot more. If your friend is going to write his own grants, it is true that a lot of grants do not accept proposal from non-US citizens. Try to scout around some grant websites such as Grants.gov and do a search. The site allows you to screen out grants by applicant's visa status. As far as I have seen, PhD fellowship grants mostly welcome all kinds of visa status.
So, in conclusion, clarify with the said supervisor, and start assessing the grant availability.