Cool topic idea.
(Although some of the following comes from various nitpicks I have with current source ports, please don't see this as veiled criticism. Obviously I'm aware of the constraints involved, and am grateful for the work you all put into Doom. This is just what I'd personally love to have.)
First off, I'd go with a clean, nice looking UI. No Doom font. It might be iconic, but to me it just looks bad. Settings set in intuitive categories and corresponding subcategories, fancy tabs for keymapping and the like. Tooltips on each option, telling you what the option does in layman's terms.
I'd want to have strict compatibility levels (when it comes to gameplay; graphics should be entirely customisable), but some of them going all the way to advanced features seen in the most modern ports. There would be no ongoing development of new features, so the standards would stay standards.
Now the latter might prompt people to ask why, why would you go for something that sounds like a negative, so here's my angle: I feel stability in the feature set is important if you're trying to make mods centered about gameplay. When balance is finely tuned seemingly slight changes in behavior can twist everything. Also, if you're learning new features all the time, you don't have the time to really dig in and explore all the possibilities in what's already there.
There would be score in the normal game, basically point rewards you'd get for killing enemies, finding secrets and picking up items, as in Serious Sam. Also leaderboards, for both points and time. Those would come with a super special anticheat system that would somehow work perfectly even though this is a local game. ;) Of course, you could still cheat, the leaderboards would just be disabled the second you bring up the console or type a cheat code. Demos would be automatically recorded and uploaded (unless you opt out), and you could download any demo by clicking on the name of the player in the leaderboards, and watch it without restarting the game.
There'd also be an ingame content browser tied to idgames, again downloading and putting stuff in the proper folder without forcing a game restart, so you could play right away. An option to rate the mod and write a comment could pop up at the end of the levels or play session. You'd be able to write "private" comments too so you can give yourself a reminder what a particular mod is about, and the port would keep a history of your playthroughs.
Twitch.tv integration. Because why not? More mainstream exposition of Doom couldn't hurt.
The port would also keep track of your stats. Accuracy, monsters killed, favorite weapon, distance traveled, times you exploded yourself with a barrel, every unnecessary and therefore essential number you could think of.
Co-op allowing for both p2p and dedicated servers. And, uh, magical computer pixie dust to make the game as smooth in online play as most modern games feel with similar pings. I find movement and collisions in particular never feel right in online Doom compared to offline, even with 30ms. Which is weird because I'm not the most demanding person when it comes to ping, i.e. I play most games at 50-80ms without batting an eye, and L4D still feels playable at 200ms.
Hmm... So that's it, mostly convenience, fluff and personal stuff. Current source ports do a pretty good job at tackling the important things, so there's not a whole lot to ask for, even when being unrealistic.