With the exception of some newer extended-stay suite hotels, every
hotel I have stayed in has a strip of marble under the bathroom door. 
What is the purpose of this strip of marble and why is marble always
the material of choice?

One of my many jobs I've experienced throughout my life has been that
of a carpet layer. So I happen to know this one.

Apparently the use of Marble as a threshold goes back some time,
VIRGIL, mentions a marble threshold in his  AENEID

"On brazen steps the marble threshold rose, 
And brazen plates the cedar beams inclose: "

Most of the bathrooms you will see with this, have a ceramic tile
floor. Ceramic tile looks good and is durable. However the edges of
the ceramic tiles tend to chip easily. Chipped tile not only doesn't
look good, it allows moister to get into the grout area, and then all
kinds of problems begin to occur. So there needs to be something to
guard the tile edge and make a transition to other floor types such as
carpeting.

Marble transitions work very well, they are easy to keep looking clean
and new, and they add a touch of class to the transition.The don't
chip as easily as ceramic, and they give a good abutment for the
carpet to tuck against. Other benefits to this are ; No mildew, Smooth
finish, Provides a water barrier

I did a search to find some reference points for you, and came up with
these links for further reading.

Home repair and Remodeling
http://www.theglassceiling.com/h4h2/hh23_til.htm

A contractors comments on the usage of marble transitions. 
http://www.contractorszone.com/Injustices/Pikul.pdf 

Dove Carpet FAQ's
http://www.dovercarpet.com/html/q___a.html

FloorsFirst.com FAQ's
http://www.floorsfirst.com/main/Products/Ceramics/CeramicQuestions.htm


And here is a picture of a marble threshold in a Greek Royal Palace
http://www.perseus.tufts.edu/cgi-bin/image?lookup=1990.27.0623&type=site
 

Google Searches

+"Marble Threshold"
+"Marble Threshold" +Archaeology
+"Marble Transition" 

I hope you have found this answer useful, and thanks for the question.

webadept-ga

Thank you for the thorough answer.  My mind can rest now that I know
about these thresholds.

A note about the research:  When I researched this myself, I did not
know to call the marble strip a transition.  Knowing that term would
have made my search much more fruitful.

Do you mean the bottom part of the door frame? The bottom of the threshold?

Perhaps a slightly jaded comment, but as the manager of a motel (which
shall remain nameless to protect the schemers involved) I was stumped
when I received a memo stating that a switch to marble thresholds was
REQUIRED. I went and took a good look at our existing thresholds and
thought, "This is silly; as a guest I would never want to pay a higher
room rate to cover this improvement." When I asked the corporate
office why the switch was necessary the response was, "This decision
has been made to add luxury words to our brand's bathroom description.
Please begin using the phrase "marble accents" in all your descriptive
materials."
I love the historical angle and am so glad it will be a more effective
threshold, but I'm pretty sure the evil manipulators lurking in the
marketing/advertising departments are the actual motivation behind
this one. ;)