Hand out index cards (a different color each month) at the entrance. Each student who wants credit puts his or her name, school, and teacher's name on the card, then at the end of the lecture puts it in a box at the exit. (If there are several exits, have several boxes. If you're worried about box tampering, station a host at each exit if you have enough hosts. And yes, you have to bin the cards by school and teacher afterward.)

This is, indeed, essentially the solution I had considered myself. I feel as though I still like it the best even after giving due consideration to your long list of ideas, which did include several ideas I had not thought of. (For instance, I really like the stamp idea; unfortunately our lectures are Saturday mornings at 10 a.m.) I like the cards because they penalize both late arrival and early departure. (Whereas putting the sign-up before the talk only reverses the problem.) And it makes it challenging to slip in the names of students who are not in attendance, because each student receives only one card.

Some issues and possible solutions, for further consideration:

• Students will not know what to do with an index card handed to them at the door.
  • Possible solution: Use preprinted slips with labeled blanks for name, teacher, and school rather than blank cards.
  • Possible solution: Large instructions on the board indicating what to do.
• Many students will not have a pen or pencil.
  • Possible solution: A box of 144 golf pencils looks to be very cheap.
• It is more inconvenient to photocopy the attendance records before mailing them back when the names are all on individual cards.
• Anyone who leaves the room and comes back in again (e.g. to go to the bathroom) before the lecture starts can get a second card.

And two things which are not problems individually, but are sort of tricky in combination:

• There's an obvious exploit if the cards are identical every month. This is the reason you suggested different colors.
• Requires a sufficient surplus of cards that we don't run out if attendance one month is much higher than average.
  • Not a problem by itself, but combined with the necessity of making them different every month, this leads to a lot of waste, since last month's leftovers can't be reused.