Use the first one. "She look so pretty that Ian fainted at the sight of her". The "at the sight of him/her/them" is an example of a fixed expression. This one meaning that she was either so beautiful or terrifying that Ian fainted. An example of "his/her sight" which has a different meaning would be, "Jeremy was so angry he wanted everyone out of his sight". Meaning that Jeremy didn't want to see anyone.

"Sight", like "looks", is a confusing word in that it can point in two possible directions: the person looking or the person being looked at. Context almost always makes it clear in which direction these words point so you can use them without worrying too much. #1 and #2 are both fine. #1 is crystal clear and there is no confusion in #2 because of context. Everybody understands instantly what makes poor Ian faint. On the other hand, everybody will instantly understand in the following that the seeing goes in the opposite direction: "Ian was so scared that he fainted at her sight of him" in which "her sight" refers to what she saw, not what he saw.

I agree with Alex. Use the first one. Both are acceptable... the thought it clear either way, but the 'form' of the first is more proper.