The Danza: Carolyn Campbell as Carolyn Williamson.
Dueling Movies: Alien Resurrection competed with Deep Rising. They came out from within a year of each other, and featured very similar plots.
Enforced Method Acting: Winona Ryder nearly drowned when she was a child, and the first scene they shot was the underwater through the canteen scene. Winona Rider had never been in water since her accident. The looks of anxiety before she goes in, and the utter terror on her face when they can't get out the other end? Yeah, those weren't faked. She had an anxiety attack on her first day of filming.
Executive Meddling: Note that, Joss's complaints to the contrary, the script to the fourth movie is substantially the same as the one he handed over (except for the ending, as he wanted the conclusion to be set on Earth but Fox instead pushed for the one still on the Auriga). It's the rest of the project that got screwed up.
Spiritual Successor: Firefly, as pointed out by Michael Marano in the essay River Tam and the Weaponized Women of the Whedonverse. That work and Resurrection share many character archetypes and plot elements (notably the "Weaponized Woman", in the form of Clone!Ripley and River Tam.) It's evident that Joss Whedon had his work on this film in the back of his mind while he was working on Firefly.
Throw It In: Sigourney Weaver actually made that behind-the-back-without-looking basketball shot for real (and in one take, no less), and the shock and surprise the other actors show is genuine. Ron Perlman notably looked directly into the camera and said "Oh my God!"; they cut the audio for the final movie.
What Could Have Been: The climax would have actually taken place on Earth, and instead of the Newborn there was a massive white-colored Alien. Joss Whedon wrote five drafts, but Fox didn't want them so they wrote the eventual third act without his input.