In most "screened" situations (airports, schools, courtrooms, etc.) you're going to be indoors and in a crowd. Based on what I have been through, I carry the following:
1) Water bottle - 9 hours on a train without air conditioning.
2) Iodine tablets - Wish I had carried these on a delayed flight. Bathroom sink water was NOT potable.
3) Flashlight - Stuck in an elevator.
4) FAK - self explanatory. Includes safety pins.
5) OTC medications (asprin, pepto, etc)-self explanatory.
6) Spare perscription medications - lost luggage or items stolen or broken in transit. This was a problem prior to TSA coming into power. Still is, but now you know who to blame.

Other generally handy stuff for jury rigging things.
7) Duct tape - Used mainly for shoe and clothing repairs. Mini-rolls are sold by backpacking outfitters so you don't have to lug around the "Home Depot" sized rolls.
8) Nylon Cord - Same. Also, can be used to decorate backpacks, keychains, etc.