If we're going to have an extra day, put it in the summer when the weather is nicer. Make it a holiday, too–since salaries are based on a year's work, the extra day arguably means you work one day more without extra pay during leap year (so, following the normal practice, that means this year the holiday would have been observed on Friday).
If you look at Appendix D to The Return of the King, you’ll see the hobbits got it right–they set up their calendar so every year had the same calendar, that is, every year started on the same day as the year before and the months did as well. This was done by having Midyear's Day as a "blank day"–a day not assigned to a weekday. The "Overlithe," the extra day added in leap year, was likewise a "blank day" and was considered a special holiday. Problem with all that is, it would never be accepted in our world because people who take the book of Genesis literally would object to any day that’s not part of a seven-day weekly cycle. There’s one proposed calendar that would resolve that problem by shortening the year to 364 days and then adding a seven-day "leap week" every five or six years (I don’t know how the calculation works to determine how often it’s needed).