Duke Nukem was named by Todd Replogle, who was trying to come up with an idea for his forth game for Apogee (after Caves of Thor, Monuments of Mars, and Dark Ages).
Todd, who at the time lived in California (while I was in Texas), always sent new game builds to me modem-to-modem, typically with Y-modem or Z-modem protocal, whichever we could get working. (We even used X-modem once in a while, but it was dog slow, though the most reliable.)
Anyway, uploaded to me a short graphics demo that had this crude person jumping around. Todd's name for the game was something like Metal Future (though that wasn't it exactly -- I can't remember the exact name). I asked him if he had a character name and he said Duke Nukem. I immediately loved the name and told hem that that should be the game's name. I also defined the character for Todd based on what I thought the name Duke Nukem conveyed, and then I found two other artists to dramatically redraw the character:  Jim Norwood, who later created Bio Menace on his own, and then was the lead designer on Shadow Warrior, and  George Broussard, before I asked him to join Apogee as my partner -- Yes, George used to be a halfway decent artist.
For the original Duke Nukem game I created all the levels in the first episode (the shareware episode), and all the game's sound effects. Allen Blum, our current most senior level designer (also the lead mapper on Duke Nukem 3D), and now working on DNF, also played a big role in creating the first Duke Nukem game, having written the game's editor tool, and designing most of the non-shareware levels.
So, the short answer is that Todd came up with the name, but I tried to make Duke into a cool character and made him the focus of the game.