The Star Trek Encyclopedia (3rd ed., p. 471) stated the ship was "named for the noted Russian space explorer."
The ship's name was an homage to the original series character Pavel Chekov. This starship (named Chekhov, after the Russian writer, in the episode script) was originally to be mentioned by name in dialogue in "The Best of Both Worlds", but the writers ultimately felt that the debris field that remained of Humanity's apparent last stand was no place for such a "cute" reference, and the USS Tolstoy was dubbed in during post-production to replace it. The name of this ship was derived from Ed Miarecki, the creator of the design study and filming model of the Chekov. 
Its registry, according to the model featured at Ex Astris Scientia, was NCC-57302. The Star Trek Encyclopedia (3rd ed., p. 471), on the other hand, labeled the registry as NCC-53702, which could be a typographical error as it appeared to differ from the studio model used for the visual effects of the battlefield. Photographs of this model were part of Michael Okuda's art department slide show given at conventions throughout the 1990s.