The Science we are Supporting
When asked about the kinds of science being supported by the AAVSO and it's GRB Transient Search community, Arne Henden offered the following insight:

1) Few afterglows have been seen.  We need a larger sample, and
  HETE should double the known afterglows in its first year.  The
  ones we have seen have different power-law decays, no seeming
  correlation between gamma-ray brightness, afterglow brightness,
  and redshift.  Correlations may be found with the larger sample.
  The hope is that GRBs will eventually become a 'standard candle'
  and can calibrate distances to the edge of the universe.

2) There should be several classes of gamma-ray bursts.  Even in
  the BATSE catalog, there are short bursts and long bursts, with
  perhaps an intermediate class.  Only long bursts have detected
  afterglows so far.  Why?  Again, a large sample will help.  Or
  perhaps the short bursts have faint afterglows, so early detection
  will help.
 
3) There are differences in the models depending on beaming or
  isotropic radiation.  Early detection and photometry can distinguish
  between these.
 
4) The models suggest the prompt radiation comes from a different
  mechanism than the afterglow.  Only one burst has had its prompt
  radiation detected.  HETE should allow telescopes to get on
  a burst quickly enough to follow the prompt decay.  Here again,
  professionals can't act quickly enough to catch many of these
  prompt decays, but amateurs can (especially with their longitudinal
  spread).
 
5) Quick determination of coordinates and approximate magnitudes
  give a 'heads-up' for the professionals so that they can train
  their telescopes and get spectroscopic, photometric and polarization
  data on the bursts.  Their narrow fields of view require the
  good coordinates, and the early photometry tells them whether it
  is worth going after a given burst.  For example, bright afterglows
  can be observed spectroscopically with meter-class telescopes if
  caught in the first hour or so, thereby freeing up the big 'scopes
  to follow fainter bursts.
 
6) Early detection means photometry when the afterglow is 5-10 magnitudes
  brighter than its 24hr value.  This means you can work on afterglows
  that suffer a fair amount of extinction, whether in our galaxy or
  in the local medium of the source.  This again increases the sample size.

7) Early detection means amateurs can contribute UBVRI calibrated photometry,
  again freeing up the professionals from having to do photometry in addition
  to their other instrument investigations.  Amateurs can often do higher
  time resolution work than the bigger chips (longer readout) of the
  professional telescopes, again giving more detailed information of the
  early time history of a burst.

8) Professionals cannot follow every burst; GRB research is just one program
  at most observatories and cannot take every minute of telescope time.
  So amateur observations will help increase the sample size.
 
There are a dozen other reasons, but this is a good start.
Arne