As someone who has just completed the interview process for fellowships, I can at least speak to that process. I'll say that going to a "big name" residency helps a bit from the perspective that the faculty at those programs tend to be well-known and have a lot of connections among the other ortho departments in academia. That said, the emergence of the match process for ortho fellowships has made this slightly less important, as the "phone call and a handshake" method of getting someone a fellowship is a thing of the past.
What is probably at least equally, but probably more, important is that your residency train you well. There are programs that have a reputation for training excellent residents (not only surgically but also in terms of knowledge). Fellowship directors are likely to favor places that have historically trained people well. Unfortunately, this is a very difficult thing for a residency applicant to gauge about any particular program. That said, you can try to get a feel for it.
If the fellowship match process has done anything, though, it's made it easier for fellowship directors to place a higher emphasis on individual accomplishment during residency than what had previously been.
Hope this helps.
