I assume that one of the sources is MathWorld. But the question they claim Avez attributes to Gelfand is not the distribution of the leading digits generally, but specifically "will the digit 9 ever occur" as the leading digit in $2^n$ (the answer is yes, but the smallest $n$ is 54). They also link to Avez's 1966 book, which is their source. I was unable to access the book so far, but Eising, Radcliffe and Top in Simple Answer to Gelfand’s Question confirm that the problem is attributed in it to Gelfand, and add that Arnold-Avez's book published two years later replaced 9 by 7.
So it appears that Avez is everybody's source for the attribution to Gelfand. What about Avez? Here is from Arnold's obituary:"Arnold spent 1965 in Paris as a postdoctorate at the Sorbonne. At the request of his supervisor, J. Leray, Arnold delivered a one-semester course on dynamical systems. The audience included many renowned mathematicians (Cartan, Douady,
Fréchet, Godement, Leray, Schwarz, Serre, Thom). One of the participants, Andre
Avez, recorded the lectures and then published them as a book".
I am going to speculate that Gelfand never published the problem, that Avez got it from Arnold's 1965 lectures, and Arnold, who was a student at the Moscow State University and took classes with Gelfand, did not need a publication to get it from Gelfand directly. In particular, Arnold gave multiple talks at Gelfand's seminar in 1964-65, handwritten notes (in Russian) are available online. On the other hand, Arnold-Avez do not attribute the problem (with 7) to anybody, so either Arnold forgot, or Avez misinterpreted his French in 1965, and the problem was Arnold's own.