It seems to me that the questions most likely to be asked in any examination on Shakespeare's Hamlet would include these:
Is Hamlet really mad?
If not, why does he pretend to be mad?
Why is Hamlet so indecisive?
Why doesn't Hamlet kill Claudius with his sword when he comes upon Claudius at his prayers? (This question is of the greatest importance because it involves the main theme of the whole play, the so-called "Hamlet Problem." Why doesn't Hamlet act as he is supposed to act, as he is obliged to act, as he has sworn to act, as he keeps urging himself to act?)
I realize that these are not exactly themes, but they are closely related to themes. The questions have been asked and answered many times in eNotes, and you can get some good ideas by referring to them under the Q&A section under the topic heading of "Hamlet by William Shakespeare."
A great place to look is just at the listing of themes on the enotes study guide, it appears to be pretty comprehensive to me. It can also be relatively helpful to look at the character analysis as it will bring up a number of themes and ideas that sometimes aren't addressed in the themes section.
Many of the themes revolve around Hamlet, the ideas about death, the purpose of life, what happens after we die, etc., are all wrapped up in his soliloquies and his interactions with other people.
There are also a number of other sexual themes involved, people have looked at and interpreted the interactions between Hamlet and Gertrude and Hamlet and Ophelia and others in order to pull different themes of incest, etc., out of them.
There's no certain answer to this. However, Love, Politics, Revenge, Madness and Melancholia, Sin and Salvation (Death and all that) could be some of the themes?
Love - explore Ophelia and Hamlet, Claudius and Gertrude, etc. All relationships, whether partners or mother-son, father-son/daughter... Real love? Infatuation? so on...
Politics - between Denmark and Norway? between Denmark and England?
Revenge - Fortinbras avenging father, Hamlet avenging father and mother, Laertes avenging father and sister...
Madness and Melancholia - Hamlet's supposed madness, his depression at the beginning of the play, etc...