Uhh... I think my players are breaking into a prison...

So, my players somehow got it in their heads that there's some kind of lore in a prison in my campaign, so I think they're planning on breaking in. They have the help of a soldier within the army, and the party is made up of a rogue, a paladin, and a fighter. They might take some other NPCs, I'm not totally sure... But, in any case, I don't know how to run prison break-ins. Uh, any tips?

For starters, the way you phrased this makes it seem like there is not any actual reason for them to break in, they just decided that to be the case. You can absolutely correct their misconception and avoid wasting everyone’s time if that is the option you want to use.

On the actual question, what you are essentially running is a heist.  For any break-in encounter, the prep and planning is the same.

First, you have to put a lot of effort into your map and security systems, which must be done in advance.  You should have multiple ways into the complex.  Multiple different security mechanisms and ways to circumvent them.  Your goal in designing a heist is to give the party multiple ways to accomplish their goal - part of the fun is figuring out the puzzle of how to get in and out.

It also helps to think of the stages of running a heist.  A good way to think about it is to break it into three parts every good heist story has.

The first is the preparation and planning stage.  Here, the party should be able to do their own scouting, speaking with possible insiders, finding building records, etc.  During this stage, they can learn about some of the entries, exits, and security systems.  Note, there need to be multiple ways to get this information, and no one source should provide complete information. The party can then use this information to plan.  It is helpful to give the party a long rest after this point so prepared Spellcasters can adjust their lists.

Second stage is the infiltration, where they try and actually get in.  And third is the escape.  These are functionally very similar, but it is very likely security will be tripped at some point, making the escape stage usually more dangerous and full of tension, as they try to avoid an alerted enemy. 

Notably, unlike other types of encounters, you don’t really need a boss.  The entire affair is one big encounter that should present constant threat and challenge - you don’t need a swingy boss encounter to pull that off.  That does not mean there cannot be a boss, like the prison warden in your case, just that the boss can be avoided or otherwise dealt with through alternative methods. Options again are key to pulling off a good heist. 

There is a free adventure here on D&D Beyond called "Prisoner 13" which pretty much is about a prison break-in to get information out of a certain inmate.
You should check that out, it might give you some inspiration.

As Caerwyn said, you could tell your players the thing they're looking for isn't there, but I strongly suggest you don't do that. Instead, put something important in the prison. It doesn't have to be the thing they're looking for (but it can), it doesn't have to be something they expect (maybe they broke in to secure a piece of tactical information about the bad guy's plans but instead encounter an important royal figure who they'd thought dead) , but they've engaged with your world to the point of driving the story forward with this choice, and you should let them drive even if that means changing the story you'd planned to accommodate them. Do this, and your players will think it's SO cool. 

As far as running goes, I would start out thinking of it like a dungeon, but then within that dungeon you should put layers of defensive measures sectioning off parts. I'd have the initial entry that might involve scaling a wall or sneaking past a guard, an intermediate measure in it middle of the dungeon that might involve stealing the right key to reach the high-security wing, and then a final security measure that maybe has a pass code to decipher or a warden you need to kidnap. 

Take into account patrols, and be sure to account for multiple fail states (there's a Running the Game video about it,  but the thrust of it is basically it shouldn't be "game over" after a single failed stealth check).

Good luck! 

Some books that I've read suggest that you add some behind the screen countdown, and every time they do something that makes their stealth less, well, stealthy, you tick it down once. Treat it like a skill challenge, ask them the problem they need to overcome, how they overcome it, and what skill they're using to do so. Set a DC and let them roll. Failure ticks down the countdown, success counts as one success. Once they reach a certain number of successes, before ticking down the countdown all the way, they get where they want to go.