How do you translate "She said she thought he was handsome" into Japanese? I wonder whether it might be 「彼女は彼はハンサムだと思うと言いました。」, but I am not sure whether I should use 思う or 思っている in this context.
The OP's translation is correct.
As to 思っている vs 思う, 思う is more natural without a special context. You can think of a situation where 思っている fits better:
Kana: I don't think he is so good-looking as everybody says he is.
In this context, 思っています implies that "Whatever other people think, I think . . ."
As an aside, I propose two improvements to the original translation. First is to switch the order to
In this way you avoid the awkwardness of 「彼女は彼は . . .」.
The second is to remove the artificialness from the translation. The basic problem is that the Japanese language has a strong tendency to avoid pronouns unless really necessary. For this reason, the translation sounds artificial. A much more natural text is
Kana asked Aya what Koji looks like. ハンサムだと思うとアヤは言った。
See? You don't say 「彼女」or「彼」in a sentence like this in normal situations. In the Japanese language, you omit pronouns if obvious. That's why you don't need 彼は in the above sentence. Also you usually repeat nouns instead of resorting to pronouns. That's why the above sentence uses アヤ instead of 彼女.
You almost have it just right. You just want to switch the second は for a が because it's an inner clause.
Also, she most likely said "ハンサムだと思います" and not "ハンサムだと思っています" so that is probably correct.