In at least two episodes in Voyager’s first season, their EMH expressed desires of a ‘people’ name, as he was striving to be more than than temporary substitute for a flesh-and-blood doctor—yet later on he seems to forget all about it. Is there a reason for this (preferably a better one than shoddy writing)?
He does choose a name. Eventually. The Doctor's name is revealed in Endgame part 1, the penultimate episode of the series.
(The EMH enters with a young lovely on his arm.)
(Paris is going bald.)
EMH: Mister Paris. Voyager's pilot, medic, and occasional thorn in my side. Where have you been hiding yourself?
PARIS: I've been busy.
EMH: New holonovel?
PARIS: I'll make sure to get your input before I send it off to my publisher. Aren't you going to introduce me to your date?
EMH: Mister Paris meet Lana, my blushing bride.
PARIS: You're married?
LANA: Tomorrow is our two week anniversary.
PARIS: Well, congratulations. My invitation must have gotten lost in subspace.
EMH: Oh, you should be flattered. We took a page from your book and eloped.
LANA: Joe has a real flair for romantic gestures.
EMH: I decided I couldn't get married without a name.
PARIS: It took you thirty three years to come up with Joe?
EMH: It was Lana's grandfather's name.
Of course, this is the future timeline that Admiral Janeway comes from, and the events of Endgame may have changed the timeline so that the Doctor doesn't pick a name.
The Doctor tried several names, but is not known to have settled on one in the primary timeline.
Out of universe, his name in the early scripts was Zimmerman after Herman Zimmerman.
In several early first-season Star Trek: Voyager scripts and during the pre-production phase, the character of The Doctor was referred to by name as Doc Zimmerman after Herman Zimmerman, since the show's producers had not yet decided to leave him unnamed. (VOY Season 2 DVD trivia text version of "The 37's")
My speculation? He got used to being The Doctor. Everyone else did. There weren't any other doctors on board to get confused with. His name reminds everyone of his ability and uniqueness, which would stroke his ego just fine.
As generic as being called The Doctor seems, it became more associated with his identity than naming himself after a famous artist or someone else's relative.
The Doctor has chosen names or given names in several episodes. Some names he used only once because they reminded him of painful memories.
Despite not having an official name, The Doctor has used several names on a temporary basis. Below is a list of names he used:
Lord Schweitzer (VOY: "Heroes and Demons")
The first of a number of names chosen by The Doctor in honor of great medical figures of Earth's past. In this case, the name was used in order to interact with Freya and other characters in Harry Kim's holographic recreation of the poem Beowulf; following Freya's "death", The Doctor discontinued using the name, finding the association with her too painful to bear.
Schmullus (VOY: "Lifesigns", "Resolutions")
Given to The Doctor by Denara Pel.
Wolfgang Amadeus Mozart & Vincent van Gogh (VOY: "Before and After")
Adopted in succession by The Doctor sometime around 2378 of an alternate timeline, during a period of interest in great cultural figures of Earth's history.
Kenneth (VOY: "Real Life")
Chosen in order to interact with his holographic family.
Joe (VOY: "Endgame")
Chosen by The Doctor in 2404 of an alternate timeline, in which Voyager returned home 16 years later than it ultimately did. Apparently a permanent selection.
In addition, over the years several crew Voyager crewmembers such as Paris, Harry Kim, Chakotay, and B'Elanna Torres took to simply calling him "Doc." Kim introduced him to a holographic representation of Crell Moset as "Doc." (VOY: "Nothing Human")
Given his attitude of discarding his names everytime he has a painful or sad experience, there is no way of telling whether he will settle on a name.