It wouldn’t be perfect, but you would always know which day of the week the date was.
For example, Sundays would always fall on the first, eighth, fifteenth and twenty second of each month.
Thirteen (months) times twenty eight (days) equals 364. The extra day needed to round out the year could be celebrated as a holiday between day 364 and day 1 of the next year.
Our current calendar was introduced in 1582, over 400 years ago. I think we could do better.
What would you call the 13th month?
Undecember.
(Since the Decem in December is referring to tenth, and Undecem would be the corresponding way to refer to eleventh.)
Or rename all the months.
Or you could just speed up the rotation to eliminate that spurious day. Easy-peasy.
Smarch
The problem with 13 months is the difficulty in dividing it into half and quarters.
Unless you renamed all the months, people would be confused about when their birthday and anniversaries were.
But the real problem is that religions will never (any time soon) let us get away from a 7 day week.
I’d tend to classify that one under “Fairly-difficult-peasy”. 
Smarch is the 13th month.
But I think that might be slightly less accurate (364 days per year, when we use a 365 day calendar, which is still off by about 1/4 day a year).
12 months of 30 days each, each divided into five six-day weeks. Followed by 5 days, 5 hours, 48 minutes, and 46 seconds of partying. 
This is the International Fixed Calendar, proposed by Moses B. Cotsworth in 1902. He proposed naming the 13th month “Sol” and placing it in between June and July.
Personally, I would prefer Elisabeth Achelis’ World Calendar. 12 months. First month of each quarter has 31 days, other months all have 30 days.
The Wiki pages for each calendar list some advantages and disadvantages of each.
Be better to go to 10 36 day months, with 5 special holidays between months, 6 on leap years.
I always thought we should just do away with Tuesdays and Thursdays. A 365 day year would have 73 weeks. Work out months however you like, the extra weekends would be great! Now, I’m retired and don’t care.
Extrember
I like Tolkien’s Hobbit version; 12 months of 30 days, then 5 or 6 days that aren’t in any month, which are party days, which are also not days of the week, so all years have the same calendar.
A few years ago, my Dad, really into both photography and Tolkien, decided to take pictures of locations that looked like LoTR scenes, and assemble them into a calendar. Mum thought that sounded like a nice idea, seeing he’d got some great pictures, and was all ready to get some printed up as Christmas presents… He did it as a bleedin’ hobbit calendar, with 30 day months called things like Rethe and Winterfilth, and hobbit holidays. Mum was not happy 
Fix the existing month names while you’re at it. It’s been screwed up since Julius Caesar got his mitts on it.
Either move October, November, and December back to being the 8th, 9th, and 10th months respectively, or rename them so that their prefixes match their month number, i.e. December, Undecember, and Duodecember.
Tredecember then becomes your obvious choice for the 13th month.
While you’re at it, rename the 5th and 6th months back to Quintilis and Sextilism, or maybe Quintember and Sextember. We’re not Roman. We don’t give two hoots about Julius Caesar and Augustus.
And each week is six days. Every business is open every day, with two complete crews of employees, alternating 3 days work/3 days off.
I kind of like the month names from the French Republican Calendar.
Gary
Lousy Smarch weather.
“Don’t touch–Willie.” Good advice.
The 13th month should be called Anathema since 13 is unlucky.
Is this going to change my zodiac sign?
This is gonna really screw up my seven day pill box.