In the Soviet period, namely the 20-50-years of XX century, periodically raised the issue of renaming of Moscow to the fact to give the capital of the Soviet Union a "revolutionary" name.

So, February 23, 1927 a group of Soviet administrative staff of 216 people sent to the Chairman of the Central Executive Committee Mikhail Kalinin petition, in which he asked to rename the Moscow at city "Ilyich", arguing that Vladimir Lenin - "the founder of Free Russia" . However this request was not given an official course.

In 1938, the People's Commissar of the NKVD by NI Ezhov, citing "the opinion of the workers", proposed to rename the capital to Stalinodar, but this idea was rejected Stalin.

In 1953, after Stalin's death, was raised again the issue of renaming of Moscow - this time in the city of Stalin, but change the political situation shortly, did not give to implement this proposal.

It is not true that the Soviet Union had the habit of renaming its largest cities in order to give them a more socialist flavor. Of the top 10 most populous cities in the Soviet Union, the Soviet only renamed St Petersburg and Novonikolayevsk, cities named after Tsars (note that Stalingrad was not one of the Soviet Union's top-ten most populated cities). The Soviets renamed St. Petersburg right after Lenin's death to Leningrad, and shortly after renamed Novonikolayevsk (named after Tsar Nicholas) to Novosibirsk (now named after Siberia). Note also that St Petersburg, the capital of Russia at the time of the October Revolution of 1917, was not the historic center of Russian culture -- Tsar Peter the Great had built it in the 18th century in part so that he could have a capital closer to "Europe." The Soviets renamed neither of the historic centers of Russian power, Kiev and Moscow.

About a year after Lenin's death, the Soviets renamed Volgograd (a city named after the Volga river) "Stalingrad" to honor both Volgograd's and Stalin's role in defeating the reactionary forces (the "Whites") in the Russian Civil War (yes, you can see in this both the expanding influence of Stalin and his desire to rewrite history so that he played a prominent role in all stages of the revolution).

So the Soviets renamed cities named after Tsars, as well as Volgograd in honor of Stalin (and under Stalin, this city incorporated local towns and rapidly industrialized, though it still never approached the size of Moscow or Leningrad). These aren't reasons to think that they should have also rename Moscow. I see that /u/MeetKi has provided some examples of proposals to rename Moscow that never gained enough traction, but I have to say I find his analysis somewhat suspect -- he implies that Moscow very well may have been renamed to honor Stalin had it not been for the "changing political situation," but, first, Stalin already had a city named after him (and, by 1953, a very famous one, i.e., the site of the Battle of Stalingrad), and, second, it was not until 1956 and Khrushchev's "Secret Speech" that the political situation in the USSR turned against the cult of personality of Stalin. Surely three years was enough time to change the name of the city had they desired to do so (the Soviets renamed St Petersburg to Leningrad within a week of Lenin's death).