The importance of having a "big name" as your thesis advisor
I'm surely not the most qualified person on this site to answer, but I have been on a postdoc hiring committee in pure mathematics and I quite disagree with the existing answers. The single most important factor is going to be your letters of recommendation (with second on the list being your publications and preprints). I think having a famous advisor is going to be of minor importance in comparison; most people understand that also great mathematicians sometimes have students that are just mediocre, so that it's no great predictor of future success. In fact I heard in a sense the opposite argument being made, that if an applicant from a smaller university with a less famous advisor has strong letters from big names, then this is more impressive than if a student at a "top" university had the exact same letters.