Neyi is the accusative form of the word Ne
- Neyi beğenmedin? What haven't you liked?
- Bu arabayı beğenmedim. I did not like this car.
***There are some verbs that requires to add accusative suffix ı/i to the objects in the sentence. And beğenmek is one of them. And there are some verbs that is okay to use with or without this suffix- when you use it works like "the" in English, when you don't add, it means the object is not certain.
Çocuğu gördüm. I saw the kid.
(Bir) Çocuk gördüm. I saw a kid. ---> either you say bir or not the sentence means the same.