I have a very clever player that uses Prestidigitation for many useful things and one of the questions he asked me the other night is if he could use Prestidigitation to clean the salt out of the water. I wonder if water that has salt in it would be considered dirty. If I were to allow the cantrip to clean water in such a way am I forgetting another spell that would be tossed to the side because of this cantrip?
No, Prestidigitation only alters the temperature or flavor.
Prestidigitation is clear in its limits (PHB, 267):
You chill, warm, or flavor up to 1 cubic foot of nonliving material for 1 hour.
This is not altering the properties of the sea water, it is merely flavoring it so that you couldn't taste the salt that's actually there.
If you are trying to "clean" the salt water to make it potable, this is the section that you are trying to use is:
You instantaneously clean or soil an object no larger than 1 cubic foot.
If water is considered an object, which some do not, then you can try to apply the above aspect of Prestidigitation.
But you should still find an obstacle here in that you can't 'clean' the water of its salt. While the salt may spoil its use for potable drinking water, it does not mean the water is soiled even if you do count water as an object (which you probably shouldn't). The salt isn't something that represents being soiled, it represents something that in high doses would be a poison.
Should you want to purify that water in order to remove the salt and make it potable, you would use Purify Food and Drink (PHB, 270), which states (emphasis mine):
All nonmagical food and drink within a 5-foot-radius sphere centered on a point of your choice within range is purified and rendered free of poison and disease.
No, it cannot
Purify Food and Drink is the spell you are looking for. It is used to, among other things, "purify" water which generally means rendering it drinkable (as D&D does not have a specific definition of the word purify).
If the rules don't redefine a word, that word means whatever it normally means in everyday English.
If you ask me, it's not idiomatic to say salt water is dirty, or to use the term "cleaning" for the act of purifying it. When people talk about water that's dirty or not clean, they're usually referring to water that's not hygienic.
Ultimately the spell is open to interpretation, but I think you'll have a hard time getting people to agree that salt water is dirty.
The driving focus of the wording in 5e is to eliminate double meanings, so that what you see is what you get. Basically, spells and rules do what they say they do. Crucially, spells can't imitate the effects of other spells. Prestidigitation would make the illness taste nicer, but you would still be sick. Purify Food and Drink would be what your player is looking for. I respect the clever thinking, but unfortunately, there's no getting around using a spell slot here.
Whether or not salt water is clean or dirty is quite philosophical. Search freedictionary (not a great source, but a start) for "clean water" and it will redirect you to "drinking water." But most people are going to separate drinking water and clean water as two separate terms. I am also going to add a chemistry argument. In chemistry, "impurities are chemical substances inside a confined amount of liquid, gas, or solid, which differ from the chemical composition of the material or compound." So if the liquid we're talking about is "water" then salt would be an impurity. (Wikipedia)
When it boils down to it, the answer is "ask the DM." If the DM allows it, then yes, if not, then no. You can try to argue it, but once the DM has ruled, that's it.
Since you are the DM, you get to decide really. To you is salt water clean? Oh and the spell only lasts for one hour, so sure, you can drink it and it won't be salty, but after an hour, it will be, and your character will get sick anyway.
Yes, this can work
Whether or not this works is entirely a matter of whether or not salt water can be considered dirty water or not. If you consider salt water dirty water, Prestidigitation can clean it. If you don't, it can't.
'Dirty water' is proverbially used exclusively to refer to human refuse in some regions, but dirty water is also used frequently to refer to water that's not potable or otherwise 'gross' due to dissolved solids, for example in this unicef learning activity where it refers to water colored brown by chocolate powder.
That said, it is somewhat unusual to describe solutions with unwanted solvents as 'dirty'; 'impure' or 'contaminated' are preferred in most cases. Water, especially as regards potability, is sort of an exception in that regard.
As a result, it is reasonable both to regard salt and other contaminants in water as a part of water being 'dirty' or 'clean' and to regard cleanliness as not referring to such dissolved contaminants.
That said, there is a spell for purifying water, purify food and drink, which is made slightly less relevant if salt water can count as dirty water.
I would say "RAW", no, based on the other answers here.
But since it can warm water for up to an hour, thermodynamics says it could boil water in sufficient amounts, provided the amount of water was relatively small given the energy put in. Since it stands to reason you could begin to boil it, a clever character might be able to distill drinkable water from salt water with nothing more than the basic utilities of an adventuring sack, prestidigitation, and sufficient time.
So another answer is "Yes, but not the way your player was thinking." How you adjudicate this is largely based on the relationship (if any) between science and magic in your game, as determined by you, the DM.