Dodecember is the most common renaming of December among those who wish to rename the months of the year.
Dodecember comes from the roots dodeca-, from the Latinized form of the Greek dōdeka, meaning 'twelve', and -ember, which is a bit of a etymological mystery, but is commonly used in the construction on Latin-derived month names. This is a clear advantage over 'December', which misleadingly refers to the twelfth month with the Latin root decem, meaning 'ten'.
This word has been coined and recoined by those seeking calendar reform, whether it be a pure reform, an attempt to create a calendar for Mars or a fantasy novel, or simply a desire to regularize the language surrounding our current calendar. Very rarely the word Dodecuary appears (Google find three uses, apparently all independent coinages), but this is even more etymologically questionable, as it would indicate that it was the month of, or the month for, twelve.
Other potentially useful months include Octember/Octuary, Nontember/Nonuary, Dectober/Decuary, and Undecember/Hendecember.