Yes we/you can.
I recall seeing a famous video of a homemade version of the Cavendish torsion balance experiment from the early 1960's, made I think for the PSSC high school course. Basically, the physicist hung a torsion balance from a high ceiling by a long (>10 m?) piece of computer data tape (chosen because it would not stretch). He carefully minimized air currents. The torsion masses were two .5 kg bottles of water on a wooden bar (no magnetic interference). Mass, in the form of boxes of sand, say 20kg was piled around on the floor as static mass and then reversed in position with respect to the suspended masses. There was a clear plastic box around the balance (with a hole in its top for the suspending tape to pass through) also to minimize the effect of air currents, since the lateral force on each bottle is about G*m1*m2/r^2 = (6.7e-11)*0.5kg*20kg/(0.1m)^2 N ~ 6.7e-8 N, i.e. a lateral force on each bottle equivalent to that generated by a weight of about 7 micrograms, about that of a 1 mm^3 grain of sand. This is visible to us because the long arms of the torsion balance convert this small force into a torque on the suspending filament, and the restoring torque is itself very small.
I found an Italian dubbed version of the video on Youtube. See http://www.youtube.com/watch?v=uUGpF3h3RaM&feature=related and a slightly longer version at http://www.youtube.com/watch?v=V4hWMLjfe_M&feature=related. I believe the demonstrator was Prof. Jerrold Zacharias from MIT and the PSSC staff. If anyone can point me to the original undubbed black and white film loop, I'd appreciate it.
It looked really crude but qualitatively it worked. The mirror moved upon reversal of the mass positions. Yeah, experimental physics!! Calculate it out. Use your laser pointer. Glue mirrors. Calibrate. Give it as an experiment in class. Make a (music?) video. Put it on Youtube and embed it here. Social physics.
I also found some other do it your self experimenters with crude equipment, experimental tips (try fishing line) and different masses.
See http://funcall.blogspot.com/2009/04/lets-do-twist.html
http://www.hep.fsu.edu/~wahl/phy3802/expinfo/cavendish/cenco_grav.pdf
and http://www.fourmilab.ch/gravitation/foobar/, which uses a ladder, some cobblestones, monofilament fishing line and has videos. For the experiment in this last reference, you don't need mirrors, since you can see the balance masses move directly because their excursion is so large. See also http://www.youtube.com/watch?v=euvWU-4_B5Y
For all these experiments there is no calibration of the restoring force of the twisted filament (which Cavendish did from the free torsion period of the balance), the balance beam of one appears to be styrofoam, (so I would worry about subtle charge effects), and the beam hits the support of the fixed masses so that it bounces and we do not see the harmonic angular acceleration we might expect. This last problem is apparently well known to amateur experimenters in this field.
Another exposition and video is at http://www.juliantrubin.com/bigten/cavendishg.html
The best summary and historical exposition I found is at https://en.wikipedia.org/wiki/Torsion_bar_experiment . I did not realize that the experiment was originally designed by John Michell, a contemporary, whose designs and apparatus passed to Cavendish upon his death. See https://en.wikipedia.org/wiki/John_Michell. Newton had considered the deviation from vertical that a stationary pendulum would have near a terrestrial mountain in the Principia (1686). Although he considered the deviation too small to measure, it was measured 52 years later at Chimborazo, Ecuador in 1738, which was the first experiment showing that the Earth was not hollow, apparently a live hypothesis at the time. The same experiment was repeated in Scotland in 1774. See https://en.wikipedia.org/wiki/Schiehallion_experiment . Mitchell devised the torsion balance experiment in 1783, and started construction of a torsion balance. Cavendish did his experiment in 1797-1798. To me this is all quite inspiring.
Editorial (I'll move this positive rant to meta soon) - given the obviously widely varied audience on this site, I would very much like to see more questions like this one relating to amateur or home experiments. The analysis of the data and possible sources of errors in these experiments is often subtle, and is very instructive. To have real physicists and other clever students publicly criticize some aspect of an experiment provides something that many students may never get otherwise. The social network framework will help many newcomers from different countries learn what real science is in a way that yet another dose of imperfectly understood theory never will. And it's fun too.
Best Answer
Background
You would need a very sensitive instrument to measure the daytime vs nighttime difference in g. It is not 0.006 m/s2. It is much, much smaller than that, about $6\times10^{-11}$ m/s^2.
Your 0.006 m/s2 is the gravitational acceleration toward the Sun at distance of 1 AU. The Earth as a whole is accelerating sunward at 0.006 m/s2. You cannot measure that acceleration with any local experiment, and a pendulum most certainly is a local experiment.
What you can measure is tidal gravity, but you will need a very sensitive instrument. At noon, on object on the surface of the Earth is a bit closer to the Sun than is the center of the Earth and thus the object experiences a slightly greater sunward acceleration than does the Earth as a whole. The Sun pulls the object away from the Earth at noon, decreasing the sensed value of g.
The difference between these two accelerations is the source of the tides caused by the Sun. Tidal forces are approximately a 1/r3 force. This is why the tides raised by the Moon are about twice those of the tides raised by the Sun, even though the Sun is much, much more massive than the Moon.
What about midnight? At midnight, the object is a bit further from the Sun than is the center of the Earth and thus the object experiences a slightly reduced sunward acceleration than does the Earth as a whole. The Sun pulls the Earth away from the object away at midnight, once again decreasing the sensed value of g. The difference between the daytime and nighttime is extremely small, about $6\times10^{-11}$ m/s2. Detecting that small a change would require a superconducting gravimeter.
What about sunrise or sunset? Now the tidal gravitational force points toward the center of the Earth, but with about half the magnitude of the outward tidal force at noon/midnight. The difference between the solar tidal force at sunset and at noon is measurable without needing a superconducting gravimeter; you only need to be able to measure to eight significant digits. There is a problem, however: the Moon. The lunar tides are about twice as strong as the solar tides. It might be easier to conduct your experiment when the Moon is new or full, making the lunar and solar affects additive and easier to measure.
Measurement
Measuring this won't be easy. A seconds pendulum nominally has a period of two seconds. The length of such a pendulum is a bit shy of one meter. (Aside: This was the definition of a meter recommended by French scientists. There was one problem: It was a "placist" definition because gravitational acceleration varies somewhat over the face of the Earth.) Assuming a seconds pendulum with a nominal, non-tidal period of two seconds, a seconds pendulum will have a period of 2.000000167 seconds at noon at the sub-moon point during a solar eclipse and a period of 1.999999917 seconds at sunset of the same day. An hour as measured by the clock will be 0.45 milliseconds shorter at sunset than at noon.
That's not much of a difference, and you'll have to wait until Oct 23, 2014 for a solar eclipse, and you'll have to go to central America to be close to the sub-moon point. Fortunately, the effect isn't significantly reduced at a non-eclipse new moon, and the latitudinal effects aren't terrible so long as you don't live in a far northern city such as Anchorage or Helsinki.
That 0.45 millisecond difference is measurable, and with relatively inexpensive equipment. Having students start and stop a millisecond accuracy stopwatch won't work. You'll need something more sophisticated than that. Set your physics and engineering students to the challenge.
Calculations
The two links above are the WolframAlpha calculations that result in the 2.000000167 and 1.999999917 second periods of a nominally seconds pendulum. There are some magic numbers in those calculations.
This is the length of a seconds pendulum, with g=9.80665 meters/seconds2.
This is the defined value of g.
This is the combined effect of the Sun and Moon at a solar eclipse.