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Hilbert's Hotel is a famous story about infinity attributed to David Hilbert (1862-1943).

Is it documented that Hilbert's Hotel is in fact due to Hilbert, and if yes, where?

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The True (?) Story of Hilbert's Infinite Hotel, by Helge Kragh (2014)

What is known as "Hilbert's hotel" is a story of an imaginary hotel with infinitely many rooms that illustrates the bizarre consequences of assuming an actual infinity of objects or events. Since the 1970s it has been used in a variety of arguments, some of them relating to cosmology and others to philosophy and theology. For a long time it has remained unknown whether David Hilbert actually proposed the thought experiment named after him, or whether it was merely a piece of mathematical folklore. It turns out that Hilbert introduced his hotel in a lecture of January 1924, but without publishing it. The counter-intuitive hotel only became better known in 1947, when George Gamow described it in a book, and it took nearly three more decades until it attracted wide interest in scientific, philosophical, and theological contexts. The paper outlines the origin and early history of Hilbert's hotel paradox. At the same time it retracts the author's earlier conclusion that the paradox was originally due to Gamow.

The relevant quote from Hilbert's 1924 lecture is as follows (my translation from German):

An application of this fact is provided by the hotel manager, who has a hotel with a finite number of rooms. All these rooms are occupied by guests. When the guests exchange their room in any way, so that again no room has more than one guest, then that will not free a room, and the hotel manager cannot in this way make space for a newly arrived guest. We can also say: A part of a finite quantity is never equal in number to the whole. [...]

How is this for an infinite quantity? Let us take as the simplest example the quantity of integer numbers. Here already this law "a part is smaller than the whole" no longer holds. We can explain this important fact easily using our example of the occupied hotel. This time we assume that the hotel has infinitely many numbered rooms, $1,2,3,4,5,\ldots$, in each of which there lives a guest. When a new guest arrives, all the manager has to do is to allow each of the old guests to occupy the room having one number higher, and this will free room number 1 for the new arrival. Of course, in this way space can be made for any finite number of new guests, and in this world of an infinite number of houses and occupants there will be no housing shortage. [...]

Indeed, it is even possible to make space for an infinite number of new guests. For example, each of the old guests, originally occupying room number $n$, just has to move to number $2n$. Then the infinite number of rooms with odd numbers become free for new guests.

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