You have probably learned tető elsewhere in the course as "roof", but it can also mean "top" (the top of something) and in names of mountains it is basically the same as the word "Peak" in English.
You're seeing it here with the -n ending meaning "on".
I think, in relative terms, square mile for square mile, Kékestető is a huge mountain. The USA is about 106 times as big as Hungary. Sooooo.... I'll be generous. I'll accept a 60-mile-high mountain... :)
Kékes is the name of the mountain, and Kékestető is a village on the top of the mountain Kékes, now Kékestető is a part of town Gyöngyös.
Should "What is there on Kekes?" be accepted as well? (It was rejected but I'm not sure whether to report it.)
You usually don't use articles in English in conjunction with proper nouns. But then there's the question whether "Kékes" should be treated as a proper noun. I think it should be, since "Kékes" doesn't mean anything in English.
You do see definite articles used in a lot of such cases, though... The Matterhorn, The Zugspitze, The Grand Canyon, The Dead Sea, The Arctic Ocean, The Gulf of Oman... any one of those would sound wrong without the "The" in front of it, in most normal contexts. "The Kékes" feels like it's in the same category to me, and that's how I would refer to it aloud in English.
"What is on top of the Kékes." means "Mi van a Kékes tetején?" (There is a small village on top of the Kékes, its name is Kékestető. - Egy kis falu van a Kékes tetején, a neve Kékestető.)
The difference between "What is on Kékes" and "What is on top of Kékes" is subtle. It's debatable whether Rka's answer is acceptable or not, but I think it's worth a shot for her to report it.
The main problem is that the above Hungarian question - "Mi van a Kékestetőn?" - is wrong, it doesn't make sense, because Kékestető is a village. "What is on a/the village?" it is completely pointless, this causes the confusion. You have to report the Hungarian version of the above question, not the English one.
I think the most natural way to say this in English is "What is at the top/peak of Kékes?" "On top of" describes a spatial relationship, but that's obvious when talking about a village on the peak of a mountain. To me it sounds a little silly to say that the village is "on top of" the peak. Yes, we knew that - where else would it be? Underneath the peak? :) "At the peak" implies on top of it, IMO.