Saved for the Sequel

A plot element that doesn’t get fully developed, perhaps on the assumption that they’ll have the sequel to take care of it. The element can be a relationship, a theme, or a character that clearly has a lot more possible development, but doesn’t get it. The central plot is no less complete and these aspects may have just been there to enlarge the universe, but they're there and available for later use- more story to tell later.

Compare Sequel Hook, which only occurs at the end of the story and is specifically geared toward inviting a sequel.

The mouse in What Happened to the Mouse? can seem like one of these, but unlike a mouse, something that has been saved didn't disappear, it just wasn't resolved—it is still present.

Examples of Saved for the Sequel include:

Fan Works

In After Case Report, details about how the Spice Girls and the The Bella Twins along with Daniel Bryan teamed up to set a trap to trick the kidnappers into handing over information about what happened to Melanie in Case of the Missing Technology hinted at.

Film -- Animated

Po's origins in the first Kung Fu Panda. It looks like we're going to find out toward the end of the movie, but then that turns into The Un-Reveal. The actual reveal of his origins turns out to be an important part of the premise of the sequels.

Film -- Live Action

Batman Begins is an excellent example. When it ended, the plot was resolved, but there was unresolved romance between Bruce Wayne and Rachel Dawes to provide fodder for the next movie.
This is attempted with Humma Kavula in the The Hitchhiker's Guide to the Galaxy movie. The Point of View Gun was clearly setting up a plot where he tries to brainwash everyone in the galaxy and take Zaphod's place. Too bad there won't be a second movie to make these things worthwhile.
The Phantom Menace and Attack of the Clones ended with several plot points dangling to be picked up in the next installment. Technically, even the last movie of the prequel trilogy, Revenge of the Sith, ended with dangling plotlines... which were already resolved in the original Star Wars trilogy.
In Tron: Legacy, we meet the son of the bad guy from the original Tron, who's played by Cilian Murphy. Since avenging one's dead parent is a common trope, and since Murphy is a relatively well-known actor, you'd expect the son to have some relevance to the plot. But he only appears in one scene, doing nothing significant at all. Seems pretty obvious the character was in the movie only because they plan to have a bigger role for him in a possible sequel.
According to Word of God, in the American In Name Only remake of Godzilla there were going to be other Kaiju, but they got held back to appear in the sequel. Perhaps if they had put them in the movie there would actually have been a sequel.
In the Iron Man film, Rhodey looks at the silver prototype Iron Man suit and says, "Next time, baby."
In the first Christopher Reeves Superman film, General Zod is roaring to Jor-El, "You will bow down before me, Jor-El! Both you and, one day, your heirs!" However, he is then cast into the Phantom Zone and we have to wait to the next film to see him carry out that threat.

Literature

In the first Anonymous Rex book, Vincent's partner has been killed, leaving him a cynic, but how it happened is left unsaid. The second book is a Prequel where at the very end the death happens.

Video Games

During the final mission of Star Fox Assault, the Star Wolf team (who are allies to the Star Fox team for the moment) draw off the enemy ships pursuing the Star Fox team, allowing them to reach the game's Final Boss, and are not seen again, leaving it unclear whether they survived. There has been another game released in the series since then and all three members of Star Wolf did appear, confirming that they did survive, though there is ongoing debate among the fandom over whether that game is canon or not.
BlazBlue runs on this. The first game, Calamity Trigger, does not explain what exactly happened to Arakune, why Litchi is so bent on rescuing him, what Hazama is up to, who he works for, why Carl is looking for his father, what "Amaterasu" is, who the other five of the Six Heroes are, among many other plot details, creating a Mind Screw in general. Continuum Shift, the second game, answers these questions and brings up more. For instance, now that we know Carl is looking for his father because he's an absolute scumbag, we learn that he is a Chessmaster villain but nothing about what he wants or what he intends to do. Arakune is regaining sentience, but Litchi is about to do to herself what Arakune did before. Bang has the power to close portals to The Edge but has yet to have an opportunity to use it. And so forth.
In Mass Effect 2, the quarians are worried over a remote star dying far faster than it ought to be. Tali hypothesizes it's due to dark energy affecting the interior of the star, but remarks that no race has the ability or any particular motive to destroy one. Nevertheless, she voices concern over the possibility of an adversary powerful enough to take a star out. The explanation is left open for Mass Effect 3.
- Amusingly, this has ended up as an Aborted Arc - it's one of the few dangling threads that isn't addressed at all (not even in one of Shepard's emails).
Pokémon Black and White left a number of loose ends that weren't wrapped up when the story was finished. Fans were expecting a "Grey Version" following the pattern from past generations, but instead full sequels are coming out: Pokémon Black 2 and White 2.

Web Original

In the commentary track for volume 5 of RWBY, the writers coyly mention an unspecified plot element which they tried to fit into that volume, but which simply could not be done without sacrificing other necessary plot elements, and which has been pushed back to a later volume. They note that once viewers know what it is, they'll be able to see the hooks and groundwork laid for it in early V5.