Is it wrong to "rent" a Jewish person some money?
Since he needs to use it, and once he is done using it he will pay you back.
Rabbi Reisman in Hilchos Ribis deals with this. If you are going to spend the money, then that is defined as a loan and any fee for that 'rental' is 'ribis'. Renting property means that you return the same property that you rented, such as renting a tractor to plow your field. That is why there is a discussion as to borrowing flour to make a cake and when, if the price changes, returning the same amount of flour is or is not ribis. Trying to use English syntax and calling the use of the money 'rent' does not change the halacha.
@Fred points to Legendary Coins and Currency as a valid use of coins that can be rented rather than loaned. In this case, the coins themselves must be returned.
I apologize to @Fred for mistyping the citation. Thank you @DoubleAA for correcting it.
It depends on what you mean by "use," because the mode of use (both actual and contractual - see Shach YD 176:1) affects whether the money is considered "loaned" or "rented." In a loan, the money is generally meant to be used financially and may be replaced with equivalent currency. In such a case, the laws prohibiting lending on interest apply.
However, if the actual same (not merely equivalent) item of physical currency is to be returned, and if the borrower is using it for a non-financial, non-transactional purpose (e.g. displaying it for a museum show), then the money may be rented for a fee (Shulchan Aruch YD 176:1).
See The Laws of Ribbis (pp. 195-196) by R' Yisroel Reisman for more details.