he, the prince, just found out his father, the king, was killed by his uncle to take both his mother and the crown and he feels he has to face the entire political and royal machine of the country against him - and wonders if it might not be better to just die.
Unless, as I said below, that’s all in his mind. It can definitely be interpreted that way.
As far as it being named after Shakespeare’s son, it might be, or that might be coincidence, since there’s the Ur-Hamlet (a lost play which was performed before Hamlet as we know it) which may or may not have been written by Thomas Kyd. On the other hand, all we know about the Ur-Hamlet, which Shakespeare still may have written is that there is a lead character named Hamlet and there’s a ghost in it.