Welcome to the Liberator

This is the uncanny habit of a new character arriving conveniently in time to replace an outgoing one, generally in the same episode. Sometimes there is a certain logic to this, as in the case of Starfleet officers being succeeded by another of the same rank. Other times, it's just to make up the numbers, such as in Trope Namer Blake's 7, when the new character is always the same gender as the outgoing one.

When a Tabletop Game player does this in an especially cheezy fashion, it's a symptom of Complacent Gaming Syndrome.

Examples of Welcome to the Liberator include:

Literature

Kind of in the same vein as the Final Fantasy example below, in the sci-fi novel Wyrm, the main characters are very much deep into the viral MUD central to the plot when one of the characters they're playing dies. Luckily, that player just happened to have a back-up character that had been following close behind them the whole time without them noticing.

Live Action TV

In Blakes Seven Jenna is replaced by Dayna, Blake is replaced by Tarrant, and Cally is replaced by Soolin. This is the Trope Namer, in honour of the ship featured on the show, Liberator. Which was replaced by the Scorpio
In Frontline, each season features a new executive producer, all of whom are equally amoral but manipulate people in slightly different ways.
In Star Trek: Voyager, Seven of Nine is introduced in the Season 3 Cliffhanger finale. Kes leaves the show in episode 2 of Season 4 (after it's established that Seven is staying).
Averted by "The Best of Both Worlds", the Star Trek: The Next Generation season 3/4 two-parter. It looks like Shelby will be taking Riker's place as first officer, first when he's given his last shot at a command assignment, then when Riker assumes command of the Enterprise after Picard's abduction, she actually does become his first officer. Of course, Picard is saved, the ship Riker was offered command of is destroyed, and Shelby moves on to something outside the focus of the show.
In the first episode of Law and Order: Special Victims Unit's second season, Ice-T's character shows up, having been reassigned to SVU. Monique Jeffries, Michelle Hurd's outgoing character, comments, "You're here for my job."
Doyle makes a Heroic Sacrifice to save the title character of Angel. Wesley arrives in town in the next episode.
Mash... Where to begin? Trapper for BJ. Henry for Potter. Frank for Winchester. Klinger takes a more central role after Radar leaves too.
The Joel-to-Mike swap on Mystery Science Theater 3000 was a justified example: Mike's first episode is Joel's last because Mike helps Joel to escape. Upon discovering that Joel is gone and they need another test subject to continue their experiments, the Mads force Mike to take Joel's place.
In NCIS, after Agent Caitlin Todd is murdered in the season 2 finale, Ziva David appears in the two-part opening episode of season 3 - which still features Kate, after a fashion (she is appearing as other character's hallucinations of her) - but isn't revealled to be a regular until an episode or two later.
Doctor Who traditionally did this with great frequency, although there was an aversion in the last few Tenth Doctor stories. Possibly justified on the grounds that the Doctor gets lonely without a companion, or foresees one leaving, and starts actively looking for a new one.
Stargate SG-1: In the Daniel Jackson to Jonas Quinn swap, Jonas is introduced in "Meridian", which is the same episode Daniel gets sent off to a higher plane. And then, when Daniel comes back, that fits this too; Daniel's back, so Jonas skedaddles back home same episode.
In Minder, Arthur's nephew Ray arrives looking for a job in the same episode (the season opener) that Arthur learns Terry has emigrated to Australia.
Taiyou Sentai Sun Vulcan has The Hero Ryuusuke Oowashi (Vul Eagle)'s classmate, Takayuki Hiba, arrive after a hard battle. He's given a tour of the heroes' new base, and then their commander informs Oowashi that he has to go pilot a space shuttle for NASA. Of course, Hiba turns out to be there to replace him as the leader of the team.
Two and A Half Men: Ashton shows up the same episode we found out Charlie was killed.
Farscape: Sikozu for Jool (annoyingly smart redhead chick); Stark for Zhaan (ultra-mystical spiritualist); Noranti for Stark (bat-brained crazy spiritualist).
In Bonanza, Will Cartwright was set up to be this for Adam Cartwright, but it was Subverted when Will ended up leaving the show before Adam did.

Video Games

Final Fantasy IV is practically the embodiment of this trope, as another party member will appear to fill in a gap left by a previous party member leaving, typically by Heroic Sacrifice.
Near the end of Dragon Age: Origins, sparing Loghain will lead to him replacing Alistair as your weapon-and-shield Warrior when Alistair storms off in a rage.