Question:  What is the name of this clever device:
As the homes front entry door closes, a button (or rod) at the base of
the door pushes against the door jamb, and lowers a device to close
the gap between the door and the threshold to keep the weather out.

Hello.

I believe you're thinking of what's called an "automatic door bottom"
or an "automatic door sweep."

'A fairly new innovation in weather stripping is the mechanically
operated "automatic" door bottom. In this model a vinyl seal is
automatically lowered against the floor when the door is shut. The
seal retracts when the door is opened.'
http://www.leeric.lsu.edu/energy/caulking/

See:
Automatic Door sweep
http://automaticdoorsweep.com/

Automatic Door Sweep
http://www.improvementscatalog.com/home/improvements/37170-Automatic-Door-Sweep.html

Also see: Pemko Manufacturing Company Door Bottoms 
http://www.sweets.construction.com/mfg/760/P38379.htm

------
search strategy:
"door jamb" weather "Drops down"
"automatic door bottom"
"automatic door sweep"

I hope this helps. If anything is unclear, please let me know via the
"request clarification" feature.

Hello juggler:
Five stars for the five star answer.
Thank you for the answer to the “bottom of front door problem”.
My wife purchased a throw rug for our front entry area, and I need to
cut the bottom of the door for rug clearance.  And the “automatic door
bottom” is the exact answer to the problem.

Also, I checked several of your prior answers and comments to my
questions over the past several years.  We have worked together on the
purchase of a new airplane, the sale of a California residence, and
several interesting subjects from aluminum foil paper, books on logic
and drunk driving in Sweden, ect, ect.
Thank you for your disciplined, correct approach to any subject.
clicker5.  (Small tip for a cup of coffee, or a soft drink)

Clicker5,
Nice to work with you again!   Thank you for the tip and the kind words.
-juggler