What is the significance of the theme of suicide in Hamlet?

In Hamlet, the theme of suicide speaks both to Hamlet's mental struggles in the play, and the agency one has over his/her life. Depressed and ruminative, Hamlet becomes disenchanted with the promises of life. While those around him seem to do everything to preserve their lives and and success, Hamlet begins to question if life is even worth living. For the troubled prince, if one feels that life is worthless or unbearable, what should that person do? What choices do they have?When readers meet Hamlet in Act 1, he is already is depressed. He is dressed in black and mourning for his recently dead father, King Hamlet (1.2 70-72). Hamlet's sadness over his father's death is further compounded by his mother, Gertrude, who marries his father's brother, Claudius, only a month after King Hamlet's death (1.2 154-156). Hamlet is offended by this---he greatly admired his father, and to him, it seems like his mother is disrespecting him by moving on so quickly. Effectively, Hamlet feels a tremendous sense of loss; he's lost his father to death and he has lost respect and trust for his only surviving parent, his mother.
Alone and alienated, the advanced condition of Hamlet's mental troubles are quite clear when the reader first encounters him. The prince is already thinking about taking his own life. He laments:

Oh, that this too, too sullied flesh would melt,
Thaw, and resolve itself into a dew,
Or that the Everlasting had not fixed
His canon 'gainst self-slaughter! O God, God!
How weary, stale, flat, and unprofitable
Seem to me all the uses of this world! (1.2 129-134)

At the beginning of the play, the depressed Hamlet sees no value in life, and his words suggest that if God did not make a rule against "self-slaughter," he would commit suicide to free himself from his pain. Later in the play, Hamlet juxtaposes life and death, comparing them to see which is more desirable. In his famous "to be or not to be" soliloquy, he expresses the harsh nature of life (as he's experienced it) which includes oppression's power over the weak, legal injustice and unrequited love among other misfortunes (3.1 70-76). For Hamlet, life presents so much agony and pain, and he searches for reasons as to why people would choose to endure such hardships. Religion, as mentioned before is one reason, but, as Hamlet surmises, so too is the uncertainty that awaits in death, the "undiscovered country" (3.1 80).Death and suicide are clearly on Hamlet's mind, but religion, social convention and the uncertainty of death make suicide problematic for him. Life is painful and death by suicide is perceived as dishonorable/uncertain. Thus, Hamlet is not only depressed, but stuck in that depression with seemingly no avenue to escape his agony.

A theme is the central idea (or ideas) of a play. The theme of a play is the reason for the characters' motivations, actions, and interactions with other characters during the play.
Major themes that appear regularly in Shakespeare's plays include ambition, betrayal, corruption, deception, hatred, and revenge—all of which appear in Hamlet.
Although characters contemplate suicide in Shakespeare's plays (as does Hamlet), and some characters commit suicide (Brutus in Julius Caesar, Romeo and Juliet, Othello, to name a few), suicide is most often a result or culmination of the events of the play and the choices that characters make, but suicide is not a driving force for the play itself. Suicides happen in some of Shakespeare's plays, but the plays aren't about suicide.
Nobody commits suicide in Hamlet—unless you count Ophelia, who we'll get to in a moment.
Hamlet talks about suicide in the following lines:

HAMLET: O, that this too too sullied flesh would melt,
Thaw and resolve itself into a dew,
Or that the Everlasting had not fix'd
His canon 'gainst self-slaughter! [1.2.132–135]
HAMLET: To be, or not to be, that is the question:
Whether 'tis nobler in the mind to suffer
The slings and arrows of outrageous fortune(65)
Or to take arms against a sea of troubles,
And by opposing end them. [3.1.63–67]

Horatio threatens suicide at the end of the play, when Hamlet is dying:

HORATIO: . . . I am more an antique Roman than a Dane. (350)
Here's yet some liquor left. [5.2.350–351]

Horatio picks up the Gertrude's poisoned cup and starts to drink from it, but Hamlet all but knocks the cup out of Horatio's hand, because Horatio has a duty to Hamlet, and a reason to live:

HAMLET: As th'art a man,
Give me the cup. Let go! By heaven, I'll have't.
. . . If thou didst ever hold me in thy heart,
Absent thee from felicity awhile,
And in this harsh world draw thy breath in pain,
To tell my story. [5.2.352–359]

. . . but nobody actually commits suicide in Hamlet—except maybe Ophelia.
As for Ophelia, Gertrude tells the story of Ophelia's death as if it was an accidental drowning [4.7.180–198] (and as if she had nothing to do with it—but that's another story), the Gravediggers discuss whether Ophelia's death was an accident or suicide [5.1.1–21], and the Priest explains to Laertes that Ophelia's death was "doubtful" [5.1.220], and that Claudius had to intervene with the coroner to let Ophelia even be buried in sanctified ground [5.2.221–223].
There was enough doubt about whether Ophelia killed herself, or whether she killed herself because she was insane (and therefore not responsible for her own death), that Ophelia was allowed to have at least a partial Christian burial in holy ground.
Nevertheless, suicide in Hamlet is treated as an escape from overwhelming responsibilities (Hamlet must avenge his father's death), or from the harsh realities of life (Ophelia's reaction to her father's death and Hamlet's rejection of her love), or out of grief (Horatio, at Hamlet's impending death).
Whether or not suicide is a dominant theme in Hamlet, it is definitely used as a motif, which is a subject, idea, image, or event that Shakespeare explores in the play to reinforce or emphasize the main themes.

Suicide occurs throughout the story of Hamlet, with many characters committing it or intending to. Notably, Ophelia potentially committed suicide by drowning herself, and Horatio intends to commit suicide upon watching Hamlet and the others die at the climax of the play.
With so much death occurring in the play, suicide plays an important role. It is used almost as an act of defiance—where the committing party takes control of their own life and death instead of allowing another to take it from them. It is also used as a display of affection and devotion, with Horatio planning to kill himself instead of living without his leaders.
However, most of all, suicide represents the depravity inherent in the play. The house of Hamlet is so broken and depraved that characters are bringing about their own ruin, both metaphorically (with Hamlet and Laertes plotting to kill one another and others, which inevitably results in each of their own demises) and literally (Ophelia taking her own life).