Night Flight

Most Star Trek crews operate on a three-shift schedule, so that some semblance of day and night can be maintained for the staff.

Now, the idea is that the other two shifts are comprised of rotating assortments culled from all the cast you know and love already.  How boring!!!  What if the night crew was a whole new gang, toiling away without recognition, and cleaning up the messes made by that Prime Directive-Violating Day Shift?  

Here's my dream bridge crew for this "Night Court" version of Star Trek:

Commanding Officer:

Captain Alex Rogan

Even if the events of The Last Starfighter could be mashed into the Star Trek Universe, wouldn't Alex be long dead by the time Earth joined the Federation?  Ah, just have the kid circle around the solar system at relativistic speeds in a non-warp drive vessel, and you're all set!  His story arc would then be so close to "Ender's Game," Orson Scott Card would have to sue, bringing our fledgling series that "This show's been sued by a big sci-fi writer" gravitas Harlan Ellison is always dishing out.

Executive Officer:

Commander Rad-Fah-Rer

The beauty of a Quintesson second-in-command is their many-faceted managerial style:  Laughter, Rage, Bitterness, Doubt, and Death.  If this guy can't keep a team together and productive, who can?

Science Officer:

Lieutenant Cloud William

To get back to some established Trek characters:  This fellow is from a race of extremely long-lived aliens (1000 years is not an unusual lifespan) who had, generations ago, lost a culture that was exactly like our own...  Their Constitution ran (at least in part) as a word-for-word copy of ours.  As my friend Dave Shramek has noted, however, these fellows would have developed their version FIRST, maybe eons before us.  And that makes US the hapless copies.  This wasn't noticed by anyone IN the episode, but I'm assuming Spock figured it out just as the credits rolled, and "Cloud William" was plucked from the obscurity of his barbarian surroundings and started on a new career path as Elder-Sage of the Galaxy.  "Science Officer" would be something of a learn-as-you-go assignment.

Helm Officers:

Lieutenant Bele and Lieutenant Lokai

These guys would be on staff so that brawls could break out for absolutely no reason whenever the episode got a little dull.

Navigator and Communications Officer:

Ensign Pabst Fusto and Lieutenant Velveeta Mokita

"Really, man?  You're the navigator.  I know we're in the middle of an exciting space adventure, but did you really have to lose your shirt?  For the navigating?  Really?"