I been wondering about this myself, actually. It would seem that Dumbledore would leave something to Harry, but what?
Gryffindors sword, maybe. Most likely that would stay at Hogwarts.
The pensieve? He could do. At the very least Harry needs to have a look at a few more memories.
What about those whirring silver instruments that Harry was demolishing in the fifth book?
They seem interesting. I wonder what they do?
But do wizards even have wills? Have they ever been mentioned? Maybe they go by a different system.