Hamlet is so relevant to young men of today. While it's true, not many young men literally have to worry about avenging a father's murder at the hands of the man who married the widow, they do understand the tragedy of relationships breaking up and new ones starting very quickly. Young men today know what it's like to set aside dreams for someone else, as Hamlet did when he gave up returning to his studies at Gertrude's request. Many men have broken up their own relationships for the sake of pursuing something more deeply desired.
The core situation in Hamlet is whether or not to act and a consideration of what keeps him from acting. If you read the "to be or not to be soliloquy" you can make some very clear connections to our human existence. Young people (and old) ask themselves the same questions that Hamlet does everyday -- "What should I do?" and "What will be the consequences of my actions?" Young people today ask themselves about where they should go to school, what they should major in, whether they should join the military, and whom they should be with in terms of emotional relationships. The answers to those questions can shape a life! Hamlet concludes that in many cases, our contemplation of the results of our actions keeps us from acting. This may be a good thing or a bad thing, but it is certainly a relevant truth for people of all generations.
If you wanted a specific circumstance to discuss, you could look at how Hamlet must choose his friends. In the story, he has Horatio, Laertes and Rosencrantz and Guildenstern. He must evaluate all four of these men at different points in the play and he has to make decisions regarding each of them. Horatio is really the only person he can trust and he's very smart about figuring out what Rosencrantz and Guildenstern are doing. At the end of the play, he realizes he has done Laertes wrong and apologizes to him. All four characters are very significant in developing Hamlet's character.
Of course, most young men today (or ever) don't have to decide if they are going to kill their uncle for killing their father and marrying their mother. So young men's problems today aren't literally the same as Hamlet's.
I would look the idea that Hamlet's problem is complicated. There are so many different things that he has to consider when he is trying to decide what to do. This is why his problem (I think) is similar to those of young men today. In times past, young men had few hard decisions -- you just would go and do the work your father had done and you would live a traditional life.
Now, we have so many choices and they are difficult ones. What kind of college should I attend? What major? Will that allow me to have a good job in today's world or will my job be outsourced to some country where workers get a lower wage?
I think the real similarity comes because of the number of choices faced by Hamlet and because of how difficult those choices are.