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A player character falls into a 10 foot x 10 foot concealed pit trap, which is 20 feet deep. At the bottom of this pit is a gelatinous cube, which occupies all of the bottom 10 x 10 x 10 (width x height x depth) area of the pit.

The rules for a gelatinous cube (from the Monster Manual, page 242) indicate that when a creature enters the cube's space (by falling into it in this case), it's subject to the Engulf rules and gets disadvantage on the saving throw.

What space does a character move to if they succeed on the Engulf saving throw while falling into this pit?

The creature is still stuck in the cube, even on a successful save.

As the question mentions, the cube has this property:

Ooze Cube. The cube takes up its entire space. Other creatures can enter the space, but a creature that does so is subjected to the cube's Engulf and has disadvantage on the saving throw.

From the Gelatinous Cube's Engulf ability:

On a successful save, the creature can choose to be pushed 5 feet back or to the side of the cube.

Emphasis mine. The operative word being that the creature can choose to be pushed, but doesn't have to if they don't want to (unlikely) or are unable to (more likely). If there is no space 5 feet back or to the side of the cube, the creature can't choose to be pushed and therefore must choose not to be pushed.

And,

A creature that chooses not to be pushed suffers the consequences of a failed saving throw.

So the creature suffers the consequences of a failed saving throw.

They don't get engulfed.

They get to choose to be pushed "back", away from the cube, which in this case means on top. This could be interpreted as swimming/floating on top of the cube, or clinging to the side of the pit. I'd leave it up to the player to decide, since they're going to need to come up with an escape plan. Presumably the cube will try to engulf them again on its turn (what else is it going to do?) and they'll have to make the save again.

Don't roll for things that are impossible.

Making the save with disadvantage is hard. However, if you've decided that the pit is completely smooth with no handholds, and don't accept the idea of floating on top of a gelatinous cube, then there really is no escape and you shouldn't let the player roll the save.

They Bounce.

The text of the Engulf action of the Gelatinous Cube states that the save is DEX.

Whenever the cube enters a creature’s space, the creature must make a DC 12 Dexterity saving throw.

Because it’s a DEX save, I’d assume that the PC attempts to fall flat on their back or stomach, in an attempt to spread out their weight and keep from breaking through the surface of the gel.

You know how you can bounce a spoon on jello but easily pierce it with a fork? If the PC enters like on their side or feet-first, they don’t have enough contact area to spread out their weight, so they slide right through. If they fall face-first or on their back, they have a lot of area to spread their weight around, so they barely avoid breaking the surface tension of the gel.

This does make for a nice trap, as if the PC decides to stand up on the gel, they immediately fall into it. They’ll be forced to crawl slowly with difficulty trying to get out.