I am looking for a pill box that can be used by someone with severe
tremors.  It should open easily and dispense only the pills for that
day, with the other day's pills well contained.

Here are two rather expensive options:

http://www.epill.com/md2qa.html

http://www.epill.com/medtime.html

These devices are interesting, but way too expensive for most.  I am
looking for a low tech device that allows a person with tremors to
open only one hour's pills at once and not have them spill on the
floor because of tremors.  Thankyou for your efforts.

This one perhaps: http://www.easycart.net/cgi-bin/DiscreetInnovationsLtd./order.cgi?orderone++Organizers+1+359+1

Or perhaps this one:
http://www.sharperimage.com/us/en/catalog/productdetails.jhtml?sku=SI562BLU#tabs

The October 18th answers have real potential, thank you.