I am currently running a campaign and one of the players wanted to roll for perception to check if there was anything around them in a basic room where nothing was supposed to be present. The player rolled a natural 20. What should I do in this situation?

Tell them nothing is present?

Except in combat, a natural 20 is not some sort of super-success that guarantees more dramatic results. It's just the best you can roll, guaranteeing success on an awful lot of tasks because they're not that hard.

"You are confident that you have uncovered all the secrets this room holds."

While the above advice is the standard solution, I have run and played in games where a 20 in a room expected to be empty results in the PC finding something useful -- but that's also a game where a 1 means something bad happens.

Sometimes a room is just a room, and rolling a natural 20 confirmed that.

Though to be fair, in D&D that's still a very important piece of information when that room could have been, or could have been filled with, mimic(s). 😉

If you're determined for them to find something then personally I'd be inclined to have fun with it; maybe a piece of furniture is in remarkably good condition such that it might be worth a bit of money (so it becomes unexpected and unwieldy loot that the party now somehow needs to get out of there if they want it), maybe they just find some loose change, or maybe they find a ring with some initials engraved on it that you can tie into a story later?

For the latter case I like to keep a file with a list of possible "seed" items that I can use as extra loot and can act as hooks for side-quests. For example, I have a stash of loot including an unidentified substance hidden behind a loose brick/floorboard, or a skeleton beside a backpack with a letter in it and so-on. None of these has a quest written yet, but if I want to drop in a little curiosity, or a red-herring anywhere I've got the ideas ready, and I can develop a quest including them later.

I know a lot of games go that way but the players shouldn't be deciding whether to make a roll or not for this very reason. Dice should only be rolled when something is uncertain.

The player decides what they want there PC to do.  In this instance they might say "I look around the room looking for anying that might be of interest"

The DM narrates the results of the adventurers’ actions. If the PC is looking around an empty room with nothing of interest then the result is certain, whether they are very thorough in their search or are distracted by something they have the same result. No dice is needed the "DM says you find nothing."  It is also possible that their is something to discover but it is more investigatin than perception so the DM would call for an investigation roll.

There can be a problem with this approach however as it can lead to metagaming. If a perception check is asked for and the roll is very poor and the DM responds "You find nothing" the party should respond in the same say as if they were told that without a roll but because there was a roll the players know that the outcome was uncertain therefor there is something in the room but it wasn't found as the person making the perception check did so poorly, and might therefore say they want to search again until they roll high enough to find it.  This can be circumvented by either using passive scores for things like perception checks or having the DM make the roll behind a screen, neither of these are ideal (players like rolling dice) but the best option is for the players not to meta game, if the party find an empty room they procede on that basis even if the players know they are walking into a trap or missing out on some valuable loot.  

It's always been bewildering to me to change the contents of a room or book or whatever just because someone rolled high. Firstly, that's unnecessary improv that could have power scaling or plot issues because you're giving random info or items on the fly. And secondly, there'll be a lot of nat 20s if everyone rolls perception a lot and that could also transform into a big issue.

If it's a room or book you haven't decided the contents of, I understand conflating the rolls for examining something and your decision for whether or not that thing has objects or information. But if it's an empty room, it can just be an empty room. No amount of searching the kobold's unused storing area will spawn a Vorpal Sword if nothing's present.

Quote from BoringBard >>

It's always been bewildering to me to change the contents of a room or book or whatever just because someone rolled high.

 Most often this occurs in the case of "the DM hasn't thought about what's in the room" -- it's not that you know there's nothing in there, you just haven't put anything there.

Quote from Jegpeg>>

I know a lot of games go that way but the players shouldn't be deciding whether to make a roll or not for this very reason. Dice should only be rolled when something is uncertain.

As you say this can lead to metagaming; this is why personally I like to roll the checks even when they're not strictly necessary, especially for searching rooms and such, as having it sometimes find something and sometimes there's just nothing helps to keep players both on their toes, and yet simultaneously surprised, because sometimes they're expecting it to be nothing.

For example for keeping watch, I'd have them roll Perception, even if I've no ambushes prepared, because I don't want them to know I don't have anything prepared. Plus extraneous rolls can sometimes lead to surprising outcomes; maybe a bad roll means a character has fallen asleep and the next player to take watch finds them muttering in their sleep, or a high roll sees the players spot something amusing but ultimately meaningless, like a wildcat trying to grab a bird, missing, and falling in a puddle before slinking away.

There's a definite balance to be found between too much rolling and not enough, and I for one definitely don't get it right; I'm constantly trying to find ways to make more checks useful, or inviting players to tell me what skill they'd like to use if they can justify it etc., but that's getting a bit off topic.

I don't mind players rolling Investigation or Perception themselves because they want to search, as it can save time so long as they don't try to roll over and over; basically my rule is once per room unless they specifically choose to examine something else that I have mentioned is in there. In general I'll either treat the roll as an instant "what do you find quickly" roll, or I'll treat it as a timing roll, i.e- the better the roll, the quicker you find the thing I need you to know about (and this determines who finds when the whole group is looking, or who finds what if I have several possibilities) etc. Searching is definitely something that comes up often enough that it's tricky to keep it interesting and varied.

There's always the classic joke response, something to the effect of: "You gain a strange, fleeting feeling that the world has gone flat; that you, everyone you know, and everything you care about are merely words on sheets of paper written by beings you can neither see, nor comprehend. You feel the presence of these beings, gathered around a table, the papers of your world laid out before them for some inscrutable purpose... and then the feeling passes, and you are back in the room. You find nothing of note."

First of all, I do subscribe to the idea that all rolls should be called for by the DM (possibly in response to something the player asked to do, but they should phrase it as an action, not a roll).

However, players can still ask to search any room, and only calling for perception rolls in rooms with something to find makes it obvious there's something to find and affects player behavior, so that doesn't quite remove this problem. First, consider if the empty room is necessary before you include it at all (sometimes there's a reason for it, granted). Second, maybe you could try to think of something passingly interesting they might learn about the room, even if it's not useful. Third... yeah, just tell them it's empty, but they're really confident it's empty.

I like to have people find things unrelated to the particular dungeon they are in, but perhaps related to other game plot points.

Fighting a bugbear dungeon, kill everything in a room, then roll a 20 on perception?   They find a note written in goblin that says "...do not trust the vampires..."

Quote from Mdhe >>

There's always the classic joke response, something to the effect of: "You gain a strange, fleeting feeling that the world has gone flat; that you, everyone you know, and everything you care about are merely words on sheets of paper written by beings you can neither see, nor comprehend. You feel the presence of these beings, gathered around a table, the papers of your world laid out before them for some inscrutable purpose... and then the feeling passes, and you are back in the room. You find nothing of note."

 "Three of the walls in the room are perfectly normal.  But for some reason you can't shake the feeling that the room has no fourth wall."

As previously suggested have fun with it.... even if its an empty room you could have implied it was once something else... for a variety of reasons.

While a natural 20 isn't supposed to be a guaranteed success outside of combat at all tables it is nice to strive for them to get something when they roll successes so they don't feel like it was a nothing roll. That also isn't mean magically give them something... maybe they just have just some supernatural insight to something that happened in the room in the past or some other such thing... that can just add a bit of something to the game.

Granted that's just my two cents.

they get to see the table, players, and dm. for an instant. then ask them to make a wisdom save