One suggestion I received is that I prepare a solution of a known activity and probe the solution with the pH meter. The problem I see with this is that any error in solution preparation can lead us to conclude that the pH meter is faulty.
Is there any other way to test the accuracy of a pH meter? We need to make extremely accurate measurements (we are studying the activities of ionic solutes).
Also, what solution has an activity of exactly 1.0?
Your school should have a purchased buffer solutions that have been calibrated against a NIST standard. If they don't, have them purchase several in order to span the entire pH range, here's one from Sigma-Aldrich. For your calibration, pick a standard solution that is close to the range you plan to work in. Here's a couple of links with some more "tips" on how best to actually perform the calibration.
This is a good question. If you calibrate a meter and then put the electrode back into the 4 or 10 buffer (i.e. not the 7 buffer) the meter reads exactly what the meter thinks the pH of the buffer is. For example, with an operational buffer labeled pH 4.00 ± 0.02 pH at 25 °C the meter thinks the pH is 4.01 and, at 25 °C it will read 4.01 but not for long. Soon the electrode will begin to drift and you will get readings like 4.03, 4.02.4.01, 4.00. 3.99 etc. The error (relative to 4.01) may exhibit a trend over time or may stay centered around 4.01. Whatever it does, the rms error over some reasonable period of time (such as an hour or 2) is a measure of the meter's accuracy. It says nothing about the accuracy of the buffer itself and thus you cannot talk about the accuracy of the pH reading based on the accuracy of the meter alone. To estimate the accuracy of the reading you can rms the accuracy of the meter with the accuracy of the buffer. Thus if the buffer is ±0.02 (you can also buy ±0.01) and the meter rms error measured 0.03 the assumed accuracy of readings is sqrt( (0.02)^2 + (0.03)^2 ) = 0.036