First, it is worth recalling which characters die during the course of the play. There is a fairly wide range of characters to choose from:
Polonius - stabbed by Hamlet
Ophelia - drowning/suicide
Rosencrantz - killed in England
Guildenstern - killed in England
Gertrude - drinks poison
Claudius - stabbed by Hamlet with poisoned sword AND drinks poison
Laertes - stabbed by Hamlet with poisoned sword
Hamlet - scratched by Laertes with poisoned sword
There is a fair amount of prose in Hamlet: even the prince himself makes extensive use of it, so unless you are very ambitious, there is no need for the monologues to be in blank verse. Rosencrantz and Guildenstern are obvious candidates to confess matters previously unknown. There is a brilliant play by Tom Stoppard called Rosencrantz and Guildenstern Are Dead, which explores existentialist philosophy through these two minor characters. This would be a fertile source of ideas, if you can get hold of it. Rosencrantz and Guildenstern might be able to tell us about Hamlet the student in Wittenberg and reveal sides of his personality hitherto unknown. Of course, there is a lot of duplication between the two—their interchangeability is a running joke—so you may wish to choose either Rosencrantz or Guildenstern.
Ophelia's perspective would also be worth exploring, since she also knew a side of Hamlet that we do not see. Assuming that her sanity is restored in death, she might also explain what she meant by some of her final remarks. Gertrude's confession would be similarly interesting. Did she know that Claudius murdered her husband? Did she suspect? Did she marry Claudius out of fear? When did she perceive the changes in her son? It would be a good exercise, whichever characters you choose, to write out all the questions you would most want to ask them before doing anything else. In this way, you can ensure that all the most important and intriguing issues are addressed in their monologues.