How can one determine what "point" on Earth is directly beneath a
given star on a given date?

Does this "point" have a correct name?

Can the point be measured with a sighting device? Extrapolated
mathematically by using the observer's position and the angle at which
the star is observed?  Or is there an almanac of some kind that could
be used?

Allowing for the rotation of the Earth, how wide is that ?point,? and
how quickly is it changing?

I am trying to formulate a riddle in which the geographic positions on
Earth located directly beneath three different stars might be
triangulated to indicate a fourth position (the solution to the
riddle.)

Thank you for your help!

don't know the answer to any of your questions except:

"...how quickly is it changing..."

depends on where you are.  if i've thought this thru correctly, as the
earth spins on its axis, every star appears to be directly over a
particular line of latitude.  the stars are so far away that i don't
think this line of latitude changes over the course of a year; the
parallax is negligible.  the only thing that changes is what point on
that line is directly under a star at any given time.

stars that appear to be directly over the poles (if there were such a
thing; even the north star is 3/4 of a degree away from being directly
above the north pole, and there is no south pole star visible to the
naked eye) would not appear to move at all, while stars directly over
the equator would appear to circumnavigate the earth every 24 hours. 
since the equator is almost 25,000 miles long, the position "under" a
particular star at the equator would be moving along the surface of
the earth at over 1000 miles per hour.

in spite of its length, this comment is mostly to act as a bookmark so
i can see when the question is answered.

-cab

The point directly overhead is the Zenith, but I don't know of any
term that applies from the star's viewpoint. The point on the sky
directly opposite the Zenith is the Nadir.

How long a star would stay at the Zenith is rather arbitrary - they're
both points, and the Zenith's moving, so you could argue that it's
only there for an instant.  If you allow some leeway, then the length
of time will depend on the latitude, as explained by barneca.  That
lack of precision in time will have an equivalent lack of precision in
the position on the Earth - not too much of a problem if the answer to
the riddle is a major city, but it's going to be a show stopper if
people are expected to dig up buried treasure!

As to how you'd do it, you'd need a good star catalogue, a good atlas,
a calculator and Jean Meeus' excellent book "Astronomical Formulae for
Calculators".

Ian G.