When did January 1st become New Year's Day?

A more complex question that it first appears. January was first declared the beginning of the year by Julius Caesar in 46 BCE. This New Year Day to celebrated the God of Janus - whose two faces looked back into the past and forward into the future.

Religious holiday?

Like many Pagan holidays (e.g. Christmas/Midwinter Festival) it slowly became incorporated into the Christian calendar, though from a Christian perspective the 12th Day of Christmas (January 6) might be the more logical transformation point from old into new. 

Throughout the medieval period there was sporadic resistance to what was generally thought to be an alien, pagan anniversary.

Calendar clash
A desire to standardise calendars eventually made NYD  the official beginning of the year. In 1066 William the Conqueror declared that January 1st a holiday. In the same 'I'll show you who's boss' spirit Pope Gregory XIII used the 1578 New Year's Day celebration to  decree 
that all Roman Jews, under pain of death, must listen attentively to the compulsory Catholic conversion sermon given in Roman synagogues after Friday night services. 
The Protestant societies of northern Europe refused to accept Gregory's decree, officially confirmed in 1582. They stuck doggedly with March 1st as their official start of the year until the late Eighteenth Century.

So New Year's Day does not have a glorious history. It an artificially imposed holiday, often used as a pretext for spot of forced religious conversion. 

New Year's Eve, however, offers more romantic associations - see Billy Wilder's superb The Apartment for example or When Harry Met Sally, which used the standard What are you doing New Year's Eve? to such good effect.

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