According to the Star Trek Encyclopedia and The Authorized Biography of Gene Roddenberry by David Alexander (page 536): 
The name "Q" was chosen by Gene Roddenberry, in honor of an English Star Trek fan, Janet Quarton. She was the first president of the UK Star Trek fan club, and Gene Roddenberry and many others spent time at her home, in the Scottish highlands.
Now, the very first appearance of another Q other than the Q portrayed by De Lancie was in Deja Q which was produced roughly a year before Roddenberry died. This situation left little room for ambiguity so I guess calling that second one also Q was a mere joke, while Roddyberry was still in charge:
Q2 was never referred to as such except in the script; Bernsen also referred to himself as such in the Star Trek 25th Anniversary Special. In the episode, both he and John De Lancie's character addressed each other as "Q".
The next appearance of a new, proper, Q (not counting Amanda Rogers) happened way after Roddenberry's death (in the VOY era) and it is not a stretch to assume that the writers wanted to keep Roddenberry's spirit by extrapolating from the previous encounters with other Q: Since all previous "proper" Q (again, not counting Amanda Rogers) were called alike and their species was referred to as "The Q" (plural) and "The Q continuum" in "Deja Q", calling the next member of the continuum also Q (the one that was later called Quinn, after becoming human) seems like a nod to the Roddenberry era. But this is again speculation on my side.
A different interpretation for calling the Quinn-Q also Q (initially) might have been a way to have the Voyager crew immediately recognise the situation they were in. This was Janeway's very first encounter with a Q and she was only familiar with Picard's log entries. This seems to be a rather convenient (and time saving) way to communicate to the audience who that person was and what was to be expected.
After the Quinn-Q, they pretty much established that all Q share the same name and stuck with it for future encounters with other Q (like DeLancie-Q's wife, Junior Q, etc.).
If we consider the in-universe aspects, you could assume the Q continuum to be so intrinsically dependant on each individual Q (as portrayed in Death Wish) that albeit still being individuals, the name Q merely refers to being part of the collection rather than being a personal name. This would be consistent with the Q civil war erupting after and because of removing one single Q from the continuum.