I've supervised a ton of internships, and was an academic before that.
The success of internships is largely preordained by the quality of the mentor. Specifically: careful project selection, good advising particularly in the first few weeks, and political/social capital to translate internships into offers (both at the company and elsewhere). Interns can screw things up, and in rare cases heroically rescue an internship, but the liklihood of a good outcome for the average intern is often preordained.
So the real advice is to find an excellent mentor. But that's really hard because you rarely know who the mentor is when applying. In lieu of better information, FAANGs and Better-than-FAANG (eg Jane Street) are a good proxy because they disproportionately pull individual contributors with significant experience mentoring juniors: PhDs, people who TA'd during undergraduate, etc.
Out of curiosity, which companies other than Jane Street would you consider "Better-than-FAANG"? I am asking because I am interested in internships, too, as I am currently an undergrad (in math).
The success of internships is largely preordained by the quality of the mentor. Specifically: careful project selection, good advising particularly in the first few weeks, and political/social capital to translate internships into offers (both at the company and elsewhere). Interns can screw things up, and in rare cases heroically rescue an internship, but the liklihood of a good outcome for the average intern is often preordained.
So the real advice is to find an excellent mentor. But that's really hard because you rarely know who the mentor is when applying. In lieu of better information, FAANGs and Better-than-FAANG (eg Jane Street) are a good proxy because they disproportionately pull individual contributors with significant experience mentoring juniors: PhDs, people who TA'd during undergraduate, etc.