What is the plus/minus variation of the solar constant? In other
words, how constant is it? How many years has it been measured using
current, agreed-upon,
reproducible procedures?

Hello knowitall22

“The term "solar constant" is a misnomer because it is not a constant.
The orbit of the Earth is an ellipse and therefore, the Sun-Earth
distance is variable. In Winter, Earth is slightly closer to the Sun (
perihelion passage occurs around January 3 ). The solar constant is in
fact defined for a mean value of the Sun-Earth distance. Moreover, the
energy radiated by the sun has changed, at least over a large
timescale. It was smaller ( by about 30% ) in the early years of the
sun's life.”
“Measuring the Solar Constant”, European Association for Astronomy
Education: http://www.amtsgym-sdbg.dk/as/solarconstant/

According to D.Crommelynck and S.Dewitte
“the observed variability observed over more than 14 years is the
response of the outer Solar layers, the photosphere in particular, to
some excitation originating somewhere near the bottom of the Solar
convection zone also responsible for the Solar spots and the
correlated photospheric features.”
D.Crommelynck and S.Dewitte, 'Solar constant temporal and frequency
characteristics',Solar Physics, Vol. 173, N0. 1, pp. 177-191, June
1997.
http://estirm2.oma.be/solarconstant/articles/article4.html 

According to data from the Jet Propulsion Laboratory of the California
Institute of Technology (“Thermal Environments” JPL D-8160): The
variation in solar constant is “approximately +3.5% about the mean
value of 1367.5 w/m2” http://www.tak2000.com/data/planets/earth.htm

You can see a chart of series of measurements of the solar irradiance
taken by the Diarad radiometer on SOHO between January 1996 and March
2002 at http://remotesensing.oma.be/solarconstant/virgo/diarad_data.htm
The values range from  c.1365.0 to c. 1368.2 W/m2 
Data are also given without the ageing correction at
http://remotesensing.oma.be/solarconstant/virgo/diarad_data_level1.htm
This is part of the VIRGO program (Variability of solar IRradiance and
Gravity Oscillations) which has a web site at 
http://virgo.so.estec.esa.nl/
NASA figures for four separate data sets give the following ranges:
    ACRIM I 1364.48 to 1369.71 
    ACRIM II 1363.75 to 1367.14 
    ERB 1368.50 to 1374.80 
    ERBS 1363.10 to 1367.60
http://daac.gsfc.nasa.gov/CAMPAIGN_DOCS/FTP_SITE/INT_DIS/readmes/sol_irrad.html


“Total Solar Irradiance (TSI) has been monitored with absolute
radiometers since 1978 onboard five different satellites:
Hickey-Frieden (HF) radiometer onboard the NIMBUS-7 spacecraft:
16.11.78 - 24.01.93
ACRIM I onboard the Solar Maximum Mission (SMM) spacecraft: 14.02.80 -
01.06.89
Absolute radiometer onboard the Earth Radiation Budget Satellite
(ERBS): since 25.10.84
ACRIM II onboard the Upper Atmosphere Research Satellite (UARS): since
October 1991
DIARAD and PMO6-V absolute radiometers of VIRGO onboard the SOHO
spacecraft: since 18.01.96

Each time series was corrected for sensitivity changes and operational
influences as described in Fröhlich and Lean (1998)”

The reference is to Fröhlich, C. and Lean, J.: 1998, Total solar
irradiance variations: The construction of a composite and its
comparison with models, in F. L. Deubner (ed.), IAU Symposium 185: New
Eyes to See Inside the Sun and Stars, Kluwer Academic Publ.,
Dortrecht, The Netherlands, pp. 89-102, 1998.
Information from “Solar Constant”, World Radiation Center observatory
at Davos, Switzerland.
http:// www.pmodwrc.ch/solar_const/solar_const.html (I was unable to
access this URL and used the Google cache at
://www.google.com/search?hl=en&lr=&ie=UTF-8&oe=UTF-8&safe=off&q=%22solar+constant%22&btnG=Google+Search
)

“The Space Absolute Radiometric Reference (SARR) was introduced after
the ATLAS 2 space shuttle flight in 1993. During this shuttle flight
several radiometers made simultaneous observations of the sun. The
intercomparison of the radiometers allowed the definition of a SARR
coefficient for every radiometer. This SARR coefficient is the
instrument adjustment coefficient by which the solar irradiance
measured by an individual instrument has to multiplied. After SARR
adjustment, all instruments measure the same mean level of the solar
irradiance.”
“The history of the Space Absolute Radiometric Reference coefficients”
http://estirm2.oma.be/solarconstant/sarr_table.html (Royal
Meteorological Institute of Belgium)

I hope I have covered this topic to your satisfaction, but please
request clarification if necessary.