The brave and the bold at CinemaBlend have been watching Lindsay Lohan's reality series. I didn't know there was such a thing. Maybe they just mean Court TV?
Anyway, whatever this show is, it featured Lohan's explanation of how she lost a role in The Avengers. She doesn't say which role but does say that it went to an unknown.
CinemaBlend try to figure out which part it might have been and, to be honest, their best guess is that it was part that later got cut out.
My best guess is that it was the waitress played by Ashley Johnson. She's no unknown in my Otis, Much Ado About Nothing and The Last of Us loving household, but I think she might qualify as one for Lohan's purposes.
Anyway, I don't know if it's fair to blame Lohan's manager for this gig not materialising. He was up against the good taste of Joss Whedon and Kevin Feige after all.