Sounds like you are doing it right.
1) Click on Edit Microbadges from your profile page
2) Click on a slot (it will highlight it with a red outline) and the page should expand to show all the MBs
3) Click on the MB you want (it should immediately move up to the red outlined box) and the screen should collapse
4) Click on "Update" at the bottom for it to take effect.
Sounds like you are not doing step 4 by your description but maybe that is because it won't let you past Step 3. If it won't let you do step 3 then it sounds like you have a bug and need to report it to admin. (I just went through and changed mine and it worked fine).
Thanks for the reply Brian. It wouldn't let me get by Step 3. It looks like it's a problem with my browser. I just used my in-law's computer and could change it with no problem. Now I've just got to figure what's wrong with my browser settings...