Shakespeare's only son was called Hamnet (which also has an entry on BtN), but a biography I read recently speculated that the name would have been pronounced "Hamlet" in Elizabethan times. So this name was clearly very emotionally charged for Shakespeare, considering that the son died in infancy -- and this was before he began to pen the play.

Shakespeare's son Hamnet died at age 11, by which time Shakespeare had penned the play 'Hamlet'.

Hamnet Shakespeare actually died in 1596, and the play was written AFTER this, circa 1601.

Not strictly name related, but I just thought I'd mention that, to me at least, Hamlet is neither 'snivelling and incompetent' nor a 'masterful man who would avenge his father's death'. I think it's pretty universally accepted that Hamlet is not masterful, and procrastinates his revenge for most of the play in a state of indecision. However, while an embodiment of the Elizabethan cult of melancholy, I've never found him snivelling; when acted well anyway. With Shakespeare's touch, his internal struggle and his deep depression make him deeply flawed and human, but also a character who I think the audience can feel deep sympathy for and attachment to. Considering the situation he's in especially, I think it would make a pretty unrealistic and unrelateable character who could enact revenge immediately and with no doubts or troubles, as well as a very short play. Hamlet isn't like the usual revenger from the Elizabethan stage, with the amount of doubts he suffers, but this makes him unique and endearing; as well as reflecting the complexities of Shakespeare's attitudes to revenge in his plays, shown also in Romeo and Juliet when we see revenge as something futile and destructive.

Hägar and Helga's son in the American comic strip 'Hägar the Horrible'.

Hamlet Mkhitaryan was a Soviet and Armenian football player who played striker for the Armenian national team and played most of his club career for Armenian Premier League club Ararat Yerevan and CFA 2 club ASOA Valence.

Mkhitaryan was one of the biggest stars of Ararat Yerevan in the Soviet Top League during the 1980s, scoring 46 goals for the club, and played for ASOA Valence during the peak of the team in Ligue 2. He is the father of Henrikh Mkhitaryan, a current Armenian international footballer.

In response to H. Gravy:

Hamnet Shakespeare was not named after a play or character. He was, in fact, named after his godfather Hamnet Sadler. Sometimes the name Hamnet was seen used interchangeably with the name Hamlet in medieval times and several instances are found in documents. https://books.google.com/books?id=DiJAAAAAYAAJ&pg=PA162&lpg=PA162&dq=abuse-not+puritan&source=bl&ots=RTCMOXO4W9&sig=WvrJuI_52Cv0Q1NUraiyE-6rr_k&hl=en&sa=X&ved=0ahUKEwjIorLqi8_VAhUoh1QKHYM8COcQ6AEIUzAH#v=onepage&q=abuse-not%20puritan&f=false.