The stone he put in its place looked and behaved exactly like Jon's stone, except that it wasn't. It was the Moon Stone, and it has a good memory for clever wishes.
Whenever a wish was made that includes a time limit or some other termination clause, the Moon Stone would bring it back "“ just as if it had been wished again "“ on the first night of every full moon after the wish has expired.
So if somebody made a wish that would last a week, the wish would return for one week upon every full moon from then on. A wish that was intended to last a year would return for another complete year at the next full moon, and again once that year has lapsed...