But those 365 days don't have to all fit into weeks. Take 364 days, divide those evenly into 7-day weeks, and then have the one (two if leap) new days be not part of the weeks. Call it Caturday or something. Hold a contest.
I always figured the funnest thing would just be not to give it a name, and then when people ask be like, well yesterday was Sunday and tomorrow is Monday.
But the more reasonable proposal I heard was just to call it New Years Day/Weekend, depending?
The fine article talks about that---there was Jewish resistance to the idea because they felt that the Sabbath was always the 7th day and that would get out of sync.