The name was coined in this 1982 article by Barry Simon:

The name comes from the fact that Mark Kac* has offered ten martinis to anyone who solves it. [...] Actually, Kac said "has all its gaps there", so perhaps one should solve instead the Ten Martini Problem, Strong Form (or should it be Dry Form).

[*] Marc Kac, public communication at 1981 AMS Annual Meeting.

The name was coined by Barry Simon in this 1982 article (page 487):

The Ten Martini Problem: The almost Mathieu operator has a Cantor spectrum.

The name comes from the fact that Mark Kac* has offered ten martinis to anyone who solves it. [...] Actually, Kac said "has all its gaps there", so perhaps one should solve instead:

The Ten Martini Problem: (Strong Form, or should it be Dry Form)...

[*] Marc Kac, public communication at 1981 AMS Annual Meeting.