Passive Rescue

Some people like to come to the rescue with all guns blazing. The Big Damn Heroes make their entrance as loudly as possible and The Cavalry indicate that the day is saved by their mere approach. Then there's people who prefer this trope. Rather than simply unlocking the door to the hero's cell or taking out the guards, they slip them a lockpick or weapon. Sometimes they'll be even more subtle; for example, by making sure that the hero's allies can find him and break him out or providing a distraction.

The reason for these can vary. Sometimes it's down to the character themselves; the Mysterious Backer would probably prefer subtlety, The Mole has to maintain his cover, The Rival probably won't help them too much if they can avoid it and some characters just can't do more than provide a means of escape. Other times it's because It's Personal between the character and their captor and the person helping them knows that This Is Something He's Got to Do Himself. Of course, some times it's a trap which has been set by their captor...

Supertrope to Jail Bake. See also You All Meet in a Cell and It May Help You on Your Quest. Sometimes a Video Game will use this avert an Escort Mission by having you free a captive who can escape on their own.

Examples of Passive Rescue include:

Anime and Manga

In Ghost in the Shell: Stand Alone Complex, The Major is trapped in an operating room (unable to move because she was half way through transfering to a new body) with an assassin (who impersonated the doctor and has her at her mercy). The Laughing Man turns up and helps her by giving her control of her body again (he probably falls under the Non-Action Guy varient, since despite being a competant hacker he's not shown to be very good at physical confrontations).

Film

In the James Bond film The World Is Not Enough, Bond is strapped in a torture device and the Big Bad is about to break his neck. The comic relief character (a Russian mobster) storms in looking for his (murdered) nephew and is promptly shot...he then reveals he has a cane gun and takes aim at...Bond. Or more specifically his restraints (probably because Bond being free to act would be much more helpful than a dead Russian mobster and dead Big Bad with Bond still restrained and unable to put a stop to said Big Bad's Evil Plan).

Literature

Discworld:
- In Interesting Times, when the Red Army is imprisoned, they find that not only has someone unlocked their cell, that someone has also killed all the guards, left them a pile of swords, and given them a convenient map to the Emperor's throne room. Rincewind notes (correctly) that this is too good to be true and that they should just quit while they're ahead: They're being framed for the Emperor's murder.
- In The Fifth Elephant, after Vimes has been locked in a cell (after being framed, sort of, for the attempted assassination of the Low King) he finds that the cell has been unlocked and the crossbow that was owned by the "assistant" Vetinari had sent to accompany him is hidden under his pillow. He realizes that whoever wanted him to escape wanted him to kill the guard to justify killing him (since he's to be the prey in some werewolves' hunt) and discards it.
Interesting example in the Ender's Game series; The character doing the rescuing knows that the (villainous) character he's been ordered to rescue will, inevitably, go out of his way to kill anyone who has put him in a state of weakness - so rather than just rescue him he opts instead to 'loan him a knife'. This puts him on the escapee's good side and sets himself up as his right-hand man, so that when Achilles needs him most he can betray him - by, once again, 'loaning him a knife'.
In Myth Adventures Skeeve's got drunk and carelessly showed the local guns his money. When they tried to rob him, he wished the genie (or Djinn) Calvin to help him out. Being a non-action Djinn Calvin couldn't do much in a fight, but he sobered Skeeve, so the latter didn't have to fight drunken.

Tabletop Games

Dungeons and Dragons module A4 In the Dungeons of the Slave Lords. When the PCs awaken in the cave labyrinth they find several spell scrolls that they can use in their escape. They were left with the party by their ally who infiltrated the Slave Lords' organization.
Shadowrun supplement Harlequin, adventure "Counterstroke". The PCs have been taken prisoner and are being tortured for information. If they're nice to one of the inhabitants of the prison, he'll help them by slipping them a monofilament saw they can use to cut through the bars and escape.

Video Games

In Metal Gear Solid Snake is captured. Otacon brings him some ketchup which can be used to escape by Faking the Dead and making the guard open the cell door to investigate (Otakon doesn't subdue the guard because he's...Otakon).
In one thieves guild quest in Oblivion you can convince a thief who's been captured to give you some information in exchange for a lockpick he can use to escape.
In Grandia, Lieutenant Leen accidentally-on-purpose drops the key to Justin's cell within reach when he's imprisoned at the Garlyle Forces base.
In the Adventure Game Innocent Until Caught, when Jack T. Ladd is imprisoned, a machine able to disintegrate walls is sent by the Tax Office to his jail, so that he can escape and they can use him as a pawn to fight tax evasion.
In a preorder/DLC only bonus mission in Deus Ex Human Revolution you rescue the son of a Triad boss. Since he doesn't like augmetics, you're given a mysterious device to give to him (which you might have found out about if you snooped on his emails) which allows him to use stealth camouflage to simply walk out on his own now that you've freed him.