Hamlet has three father-son pairs because he is examining the nature—and more to the point, the value—of revenge. All three sons in question—Hamlet, Fortinbras, and Laertes—lose a father to violence at the hands of another man, and each feels required by the revenge culture to avenge the death.

Laertes and Fortinbras both have a conventional response to the murder of a father: Laertes rushes home from France on hearing the news that Hamlet has killed Polonius, ready for revenge. He simply wants to act to avenge his father.

Fortinbras has to wait for the senior Hamlet to die to move against Denmark to avenge his own father's death. Rather than go to war, his father and Hamlet's father (the senior Fortinbras and senior Hamlet), decided, very unconventionally, to fight to the death in personal combat over a contested piece of land. The one left standing would get that territory for his country.The senior Hamlet won. Fortinbras apparently had to honor the agreement his father had made with the senior Hamlet over the territory, but after King Hamlet is dead, Fortinbras feels free to move ahead to reclaim the land by invading Denmark, in this way avenging his father.

Prince Hamlet is a marked contrast to both these men because he so deeply questions the revenge culture. He has often been called the first modern hero by scholars because of his interiority, and we do see his anguish in a personal way. Even after he knows Claudius has murdered his father—after he has tested the ghost's words—he hesitates to get caught up in the cycle of violence. He tries to screw up the courage to murder Claudius by invoking the image of Fortinbras, but even that does not work: Hamlet questions the morality of the loss of life it will take for Fortinbras to regain a few feet of land.

The quest for revenge leaves a stage littered with dead bodies and costs both Hamlet and Laertes their lives. By showing the price of the revenge culture the men live in, Shakespeare calls that culture into question.

Shakespeare employs three pairs of fathers and sons—Polonius and Laertes, old Fortinbras and young Fortinbras, and old King Hamlet and young Prince Hamlet—in order to provide contrasts that would highlight young Hamlet's failure to act in a timely manner. The other two pairs both have sons who, despite any flaws their fathers might have had, react swiftly and decisively to defend their fathers' honor. Horatio tells Barnardo,

Now, sir, young Fortinbras,
Of unimproved mettle hot and full,
Hath in the skirts of Norway here and there
Sharked up a list of lawless resolutes . . .
to recover of us, by strong hand
And terms compulsatory, those foresaid lands
So by his father lost. (1.1.107-116)

When old Fortinbras was defeated by old Hamlet, Denmark took lands that had once belonged to Norway and killed old Fortinbras. His son, young Fortinbras, wastes no time in attempting to retake those lands and avenge his father's defeat at any personal cost.

Further, when Laertes learns that his father has been murdered and that Polonius's death has been hushed up, his burial hastily done, Laertes returns from the continent, ready to storm the castle. Supported by a crowd who would see him crowned king, Laertes says to Claudius, "O, thou vile king, / Give me my father!" (4.5.126-127). He confronts the man he believes to be responsible for his father's death without wasting time or mincing words. Further, he tells Claudius,

That drop of blood that's calm proclaims me
bastard,
Cries "cuckold" to my father, brands the harlot
Even here between the chaste unsmirched brow
Of my true mother. (4.5.130-134)

When Claudius asks him to calm down, Laertes says that to be calm now would be like a betrayal of his father. There's no way he could be calm and be a true son. To be calm would be to proclaim that Polonius was not his real father and that his mother was a whore. In other words, he must act when his father has been dishonored.

Both of these sons contrast sharply with Hamlet. Hamlet learns (from his father's ghost) that his father has been murdered, then wastes a bunch of time being mad without any real plan. He only lands on a plan once the group of actors happens to arrive in Elsinore. Further, even when he has an opportunity to kill his uncle, he finds a reason not to take it (i.e., that Claudius is praying, and he wants Claudius to die without having opportunity to atone for his sins, just like old King Hamlet). Hamlet's inaction is thrown into even deeper relief because of these two examples of what a son is supposed to be, what filial piety is meant to look like.

Old King Hamlet kills Old King Fortinbras. Claudius kills Old King Hamlet. In attempting to kill Claudius, Hamlet kills Polonius. Laertes tries to avenge Polonius but is killed by Hamlet. Hamlet still wants to avenge his father by killing Claudius and Young Fortinbras wants to avenge his father by killing the heir (Claudius or Hamlet). Chaos. 

Hamlet poses his most famous existential question, "to be or not to be" as a decisive moment: to end his life or to tackle the disorder of Denmark by avenging his father's death. In the larger picture, Hamlet is also attempting to put a definitive end to this pattern of killing and avenging. Literally, this is about escaping this pattern of unnatural (murder) deaths. Opposing this unnatural pattern is the natural order of life which is father (and mother) to son (and daughter). Barring murder, people tend to grow old and die of natural causes. But so many characters (Old King Hamlet, Old King Fortinbras, Polonius, Laertes, Ophelia, Gertrude, Claudius, and Hamlet) die of unnatural causes and thus, too early. To die unnaturally is to upset the natural or familial order of life. And with murder, it is to upset the order of law. With a king being killed, this upsets the order of the royal line, the order of authority. Each son must deal with the loss of his father but also the loss of familial and social order. 

Paralleling the three sets of fathers and sons, Shakespeare provides three examples where this natural order is upset. Feeling so distraught at his father's death and his mother's "unnatural" (incestuous) and quick marriage to Claudius, Hamlet seeks not only revenge but to reestablish some kind of order. 

The time is out of joint--O curse spite,
That I was ever born to set it right. (I.v.188-89)

In Act One, Marcellus famously says, "Something is rotten in the state of Denmark." (I.iv.99) Something is wrong with the political "state" of Denmark (the king has been murdered) and something is wrong with the social and spiritual "state" of Denmark (the king's ghost haunts and demands Hamlet to avenge him). 

Each son (Hamlet, Laertes, and Fortinbras) tries to avenge his father out of love and duty but also to reestablish the order of these political, familial, and spiritual "states." When all is said and done and many are unnaturally killed, Fortinbras takes the throne of Denmark and proposes to begin a new state of order.