What does it mean to be a father in "Hamlet"? What does it mean to be a son?
Excellent question! There's not enough time to give it the in-depth answer it deserves, so I'll point you in three directions:
1) Revenge. Remember that nearly all the father-son relationships involve revenge. Hamlet is revenging Old Hamlet, Fortinbras revenging Old Fortinbras, and eventually, Laertes is revenging Polonius.
2) Ths points you toward the idial of "filial obligation" - things that a son must do for their father. Here's Claudius:
But you must know, your father lost a father;
That father lost, lost his, and the survivor bound
In filial obligation for some term
To do obsequious sorrow. But to persever...
'tis unmanly grief....
Unmanly grief. To be your own man is to break away from "filial obligation", to become a man rather than a son. But can you be a good son - and a man - at the same time? Where does Hamlet fit in?
3) Fatherly advice. Polonius, particularly, gives out advice as to how to behave to Laertes:
There, my blessing with thee.
And these few precepts in thy memory
See thou character....
"Be like me", the father says to son. And to revenge a father, the son has to become more like them: Hamlet has to find his physical violence (Old Hamlet was a warrior king!) and Laertes comes up with an underhand plot (just like Polonius does to watch Laertes abroad). Should the son follow the example of his father? Is a father worth revenging? That might just be the question.