For better or worse, Pi Day seems to be already fairly well established. Getting broad recognition for a second math-related holiday seems like a pretty daunting task. I'm skeptical that the (admittedly awesome) alternatives will be able to gain traction among the general public, in part because the problems you bring up about Pi Day may have deeper roots in preexisting stereotypes. On the bright side, I do think that propose a modified Pi Day provides us an opportunity holiday 'weekend' which would aim to broaden its focus and break some of these the stereotypes as people are paying attention to mathematics when they otherwise might not.
That saidyour question concerns, it'd be great if Pi Day had a broader focus than a single constant. I propose a math holiday 'weekend' around Pi Day so we can include all of mathematics while cashing in the celebrations!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: Clarified Removed a couple overly cynical remarks in the first paragraph expressing skepticism about whether alternatives would catch on with the general public.