Agreed that redoing an episode would be a waste of all the time, talent and resources involved (also illegal unless they pay for the rights to re-do the work of other writers), but if for any reason someone were to remake an episode, I would pick a crappy one. Remaking a fan favorite will not end happily for anyone (no fan production will ever come close matching or improving the original versions of The Doomsday Machine, City on the Edge of Forever or Amok Time, sorry) . Pick a failure and make it good. If possible.
It wouldn't be
illegal if there was no money involved. In fact, this is true of any fan
Trek production whether they use an existing script or an original one.
There was a local theatrical company here in Portland (Atomic Arts) that, for five years running, presented episodes of
Star Trek (one episode per summer) using the episode scripts verbatim, but reformated to work as a stage play. No laws were broken as they did not charge any admission. And it was very popular, by their fifth and final year, every show had thousands of people in attendance. They did "Amok Time," "Space Seed," "Mirror, Mirror," "Journey to Babel," and "The Trouble With Tribbles."
I think the key to using existing stories to is radically change the format, as Atomic Arts did with their
Trek in the Prk series. You could do an original episode, but maybe as a 1930's serial, or as a Steampunk fantasy, or as sock puppets, or anything, really. A slavish reproduction would be boring and an attempt to "improve" on the original would no doubt fall flat also.
I think the fan film productions that do best are those that ply new waters so to speak and write some original material. If doing actual TOS scripts, it's gotta be played for laughs.
--Alex