For better or worse, Pi Day seems to be already fairly well established. I propose a modified Pi Day holiday 'weekend' which would aim to broaden its focus and break some of the stereotypes your question concerns, while cashing in on its 'name recognition'.
March 13. 313 is a twin prime and a palindrome, so there are plenty of ways we could go with this, although I can't think of any specific activities. It might be fun to kick off the festivities with an organized Trimathlon event - that is, some sort of grand scavenger-hunting, puzzle-solving, team competition.
March 14. Pi Day. I'd like to second the recommendation of Buffon's Needle related activities for this.
March 15 happens to be Leonhard Euler's birthday. As Mensen suggests, celebrating individual mathematicians helps humanize the discipline. Perhaps a famous-mathematican costume contest? Euler's contributions in particular are another source of activities; the video game Katamari Damacy is a great one to do with exponential growth.
EDIT: Removed a couple overly cynical remarks in the first paragraph expressing skepticism about whether alternatives would catch on with the general public.