Would you tell me if both the following sentences are perfectly natural?
The one who stays after work this coming Friday will get paid twice the rate for each hour.
The one who will stay after work this coming Friday will get paid twice the rate for each hour.
I thought that the second sentence was grammatically correct and natural, but I've actually heard the first one used. If both are correct and natural, is there any difference in meaning?
They mean different things.
The first version, with the present tense, is probably more common and expresses a straight condition where there is a contingency based on an event in the future. "If you do X, then Y" (or, as you've put it, "the one who does X gets Y" - same idea).
The second version, with "will," can be understood by substituting "is willing to" for "will." So it can be viewed as something that is closer to a request. "If you'll do X, then Y" = "If you're willing to do X, then Y."
In your second example, you use the form "the one who'll do X, gets Y." This has more of the feel of a general request (and promise of compensation) than the first example, which is more of a simple statement of fact. They are very close in meaning in this instance. But in other contexts, where there is no element of willingness, "will" would sound very unnatural. For example, the following would be at least unnatural, if not wrong: "If it will rain tonight, I'm not going out."
