Mud is a new feature of the Well of Darkness expansion. This states that " a small figure must spend 2 movement points instead of 1 to move onto a mud space, or from one mud space to another... if a small figure only has 1 movement point, then it cannot move onto a mud space. " Further, jumping over a pit costs 3 movement points.

How much movement does it cost to jump over a pit into a mud space?

I was thinking that as there's an extra cost for moving into mud it'd be an extra movement point (to slide and gather yourself up in the icky mud on the other side of the pit). As I recall it, the hero that did this (he was running out of time to escape Area 2 in the Buried Alive quest so you can probably imagine the scenario here) didn't have quite enough MP for anything extra anyway, so we only charged the 3 points for jumping a pit, ignoring anything extra for moving into a mud space as we weren't sure if that was incorrect anyway. But it did raise the question at the time.

Yeah, this is yet another example of the Descent rules being terribly written. There are various special circumstances that involve changed movement costs, and more than one of them can apply to the same move, but they give us the final costs of specific examples rather than the rules for calculating them:

"If a hero or monster is aware of a pit, it may jump across it for three movement points for each space crossed." (JitD rules p.17)

"...a small figure...must spend 2 movement points instead of 1 to move onto a mud space." (WoD rules p.4)

And let's not forget how the designers answer when one question about this sort of thing comes up, as recorded in this thread :

" Q. In Descent JitD, how many movement points does it cost to move from inside a pit onto a mud space?
A. 3 movement points "

Note that they give an answer to the specific example situation instead of resolving the basic ambiguity in the rules...

I think the most plausible guess at intent is what you said, that entering a mud space adds +1 MP to the cost of the movement. But there is no actual rule that says that (or at least none that I can find), and there are other conceivable sets of rules that would generate all the same specific data points we actually have, so unless you can get FFG to dispense an official answer, you're probably just going to have to reach an agreement with your players. I suggest you do that before the next time it becomes a life-or-death question.

Personally, I'm still wondering if they really intended jumping to cost 3 x (pit spaces jumped), which makes jumping over one-space pits substantially cheaper than multi-space pits, or if they actually meant it to be 2x (pit spaces jumped) + 1 for entering the square on the far side, and confused themselves with the typical example, which (given the general quality of the rules-writing) is perfectly plausible.

From the Descent rulebook, page 17:

If a hero or monster is aware of a pit, it may jump across
the pit for three movement points for each space
crossed. Simply place the figure on the other side of the
pit after spending the movement points.

So you see that you aren't really moving into the space, but rather just being placed there.

That is how I would interpret it. Which would mean only 3 MPs.

However, when one looks at another portion of the rules...

From the Descent rulebook, page 16:

A figure
in a pit may climb out for two movement points, and
is then placed adjacent to the pit in any legal, empty
space(s) the owner chooses.

We have already established that climbing out of a pit into mud costs an extra MP, so if we are to use consistent terminology, the conclusion is actually that it costs 4 MP to jump a pit into mud.

Yep, it's poorly written.

I agree with Joram about the 3 MP to jump over a pit into the mud. First the jump PLACES the figure there. Second, realisticly (well considering how unrealistic other rules in Descent are...), if I wade through mud I'm slowed down as a move through it. If I jump into it that part of the slow down is lost because I'm airborne.

Personally if I were writing the rules I would have the slow down occur as you were LEAVING not entering. Movement rules tend to get tricky when moving to as apposed to moving from.