While I can't speak to naming conventions in Elizabethan England, I can say that the rest of your question is often asked amongst scholars, and that the short answer is, "We don't know."

We only have a very small number of documents about Shakespeare that are from his time, and none of them are personal writings. The personal diary, in which one records the events of one's day, did not come into existence until Daniel Pepys started keeping one in 1660 and it took some time to catch on. Even the personal essay, analyzing one's personal philosophy through the lens of personal anecdote, had only just been invented by Michel de Montaigne (his book Essais, published in French in 1580, and not in English until just a year or two before Shakespeare wrote the first draft of Hamlet), and did not catch hold in England until much later. Although letter writing was already an established practice, no correspondence of Shakespeare's has survived to the modern day. Shakespeare could have written and published a poem immortalizing his feelings about his son, as playwrite Benjamin Johnson did on a similar occasion, but if he did so no record of it now exists.

All this to say that there are no surviving documents where Shakespeare directly explains the reasons for his son's name, or what his relationship was to the story of Hamlet. So, as with nearly all biographical questions about Shakespeare, we can only examine the evidence and make an educated guess.

Hamlet is an old story. The first version was recorded in Latin by Saxo Grammaticus in 1514. By Shakespeare's time there were several different play versions in existence. One version, also called Hamlet and possibly written by Thomas Kyd, was already being performed London when Shakespeare first arrived there in the 1580s, and it is likely that Shakespeare was among the cast of a 1594 performance by the Chamberlain's Men. We can surmise that Shakespeare was also familiar with Francois Belleforest's 1570 version, entitled Histoires Tragiques, as it introduced plot elements having to do with Gertrude and Claudius' relationship that are present in Shakespeare's work, but not the 1580s play. Hamnet, having lived from 1585 to 1596 would have been born and named after Shakespeare was first exposed to the play, and died before Shakespeare wrote his first draft in late 1599. Spelling was not standardized at this time, and Shakespeare himself spelled his name in several different ways over the course of his life. So it is possible that Hamnet was named for a character that Shakespeare admired, and that his death helped to inspire, or at least added depth to, Shakespeare's writing.

Furthermore, we can guess that Hamlet was an important work for Shakespeare. It is his longest play. No surviving version of it was a performable length - it would likely run 4-5 hours if performed unabridged. It is also, from the available historical record, Shakespeare's most revised play. This self-indulgence, writing scenes that couldn't be used and spending time perfecting and adjusting a script that most would consider already stage-ready, seem to show an attachment to the subject matter. Whether this attachment had anything to do with his son, his own mental state, or just the love of an excellent story and complex characters is anybody's guess.