My cleaning lady was ready to wash the floor, but my mop was nowhere to be found. I went to my neighbor and borrowed her mop. An hour later, she called back and asked me to return it, since her cleaning lady had arrived. I replied that I needed it for another hour or so and I would return it to her then. She continued to insist that I return it immediately, so I did.
Q: Was I obligated to return it as soon as she wanted it, or can I use something until I am done once it is lent to me?
A: It is not clear from your question whether you informed your neighbor that you needed the mop to clean your house, or if you just asked to borrow it without specifying your intent. As we shall see, the wording of the request will affect the halachic decision.
Shulchan Aruch (C.M. 73:1) draws a distinction between loaning money and borrowing an object when the loan was for an unspecified period of time. Although the time period for a loan of money in such a case would be thirty days, when one borrows an object, the owner may take it back immediately.
The rationale for this distinction is that the purpose of a monetary loan is for spending it. The borrower must therefore be given ample time to earn the money to repay the loan. In contrast, a borrowed object is returned as is. As such, the borrower does not need time to be able to return the object (Sema 73:5) and must return it upon demand. Another explanation is that loaning money is a fulfillment of a positive commandment (see Shemos 22:24), and Chazal determined that if the borrower cannot use the money for at least thirty days, he does not sufficiently benefit from the money. However, borrowing tangible items does not involve a specific mitzvah other than chessed; therefore, Chazal did not assign a specific duration of time for the borrowing (Chikrei Lev Y.D. 3:128).
If one lent a tangible item for a fixed period of time, once the borrower has taken the item, the owner may not demand the return of his object until the end of that period (C.M. 341:1). Similarly, if the borrower asked to borrow the item for a specific task and has already taken possession of it, the owner may not take it back until the specified task is completed (C.M. 341:5,7).
Therefore, if you had specified that you wanted to borrow the mop to mop your house, you would have the right to keep it until you finish mopping all of your floors. If you simply asked to borrow the mop, the owner should have the right to take it back whenever she wishes.
Nevertheless, since one generally loans a mop with the understanding that at least one entire room will be mopped, it is comparable to asking to borrow an item for a specified period of time. The owner may only demand the return of her mop after the borrower could have used it to clean one room (Urim VeTumim 73 Urim 4).
As such, since the payment is for the one who is willing to search and find the lost object, it is certainly permitted to accept the $50 reward, like any remuneration for work (see also Teshuvos V’hanhagos 3:463).
It would seem that all opinions would agree in your case that you may accept the money. In this case, since the owner indicated that he was willing to reward the finder, what he is essentially doing is hiring the potential finder to find his lost object. Although it is a mitzvah for one who notices a lost object to return the item, there is no obligation to search for someone’s lost things.