Strictly spoken in terms of grammar, you are right. In English-speaking countries these days though, it is very common to say 'them' instead of 'him or her' in cases like this one
In my opinion, there are better english meanings for "gözüne girmek" like "to win somebody's favour", "to get on someone's good side" or "to keep in good with someone". Does "to give someone a good look" give the same meaning?
It is for me; (according to Turkish text)
"No matter what the person she dislikes does, he (they) can't win her love or her interest."
Because his mother is not subject in last sentence, Subject has to be a male (he/them).