Hmm your experiment sounds like a good idea but I think it'll be much harder than you're imagining. The resistance of wires is very low. After all, they are designed to conduct! Check out this table. 30 gage wire has a resistance of $0.1\: \Omega/\mathrm{ft}$ which is well below what a typical multimeter can read.
Also, because the resistance is so low, one very big source of error will be how well the multimeter probes are attached to the wire. You'll likely end up measuring the resistance at the point of contact as much as the resistance of the wire.
One way to help though is to measure very long wires. If you can get wire a few hundred feet long the resistance of the wire will start to be high enough that meaningful measurements can be made, despite experimental error.
This is a description of the experiment Cavendish performed at the end of the 18th Century to measure the density of the Earth:
Cavendish put two lead balls on either end of a long bar. He hung the bar at its center from a long twisted wire with known torque. Then, he placed two really massive objects at exactly identical fixed distances from the center of the torsion bar, in the plane of the torsion bar and at right angles to the bar at rest. The balls were attracted and started the wire twisting, but their inertia caused them to overshoot the equilibrium position of the wire. The bar wound up oscillating, and Cavendish measured the rate of oscillation to determine the torsion coefficient of the wire.
With this, he was able to determine the force attracting the balls to each other, which he used to set up a proportion to derive the density of the Earth. Here is a description of the experiment: http://large.stanford.edu/courses/2007/ph210/chang1/, as well as a derivation of the gravitational constant, Big G, that you can perform: http://www.school-for-champions.com/science/gravitation_cavendish_experiment.htm#.VUFS80uiKlI.
One can use the density derived by Cavendish, and the diameter of the Earth (which has been known since Eratosthenes in ancient Greece) to compute the mass of the Earth.
To find the mass of the Earth using the modern form of Newton's Law of Gravitation, you may enploy Little g, the Earth's gravitational acceleration, which is determined by dropping an object, any object, and measuring its acceleration toward the Earth. You do not have to know the mass of the Earth to measure an object's acceleration toward the Earth. Then, you plug the acceleration (9.81 m/sec^2), and the mass of the dropped object into Newton's definition of Force (F=ma), to find the force (F) that the Earth exerts (gravitational acceleration) at the height from which you dropped the object.
Now you know everything in the equation F = g * (m1*m2)/r^2, except for m2, the mass of the Earth. Solve for m2!
Although Newton did not know the magnitude of the gravitational constant (Big G), the form of his equation, which sets the force of gravity inversely proportional to the square of the distance between objects, was rapidly accepted by scientists because it agrees with the motions of the planets as measured by Keppler.
Best Answer
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.