What is the "rule of thumb" for determining the sample size?

As it happens, I just got back from a test on statistics. We covered that very question.

Here goes:
the sample size is dependent on how confident you want to be about your results. In the following formula, then Z is the confidence level. Use 1.65 for 90% confidence level, 1.96 for 95%, 2,33 for 98% and 2.58 for 99%.
You also need to know the standard deviation, which should be given. and the error allowable, for instance 3%, or 5% or such. That number goes into the formula as E.

so number n equals = (Z * standard deviation/E)^2

better

A rule of thumb is an easily learned and easily applied procedure for approximately calculating or recalling some value, or for making some determination. Compare this to heuristic, a similar concept used in mathematical discourse, or in computer science, particularly in algorithm design. See also mnemonic.
The term "rule of thumb" or similar exists in many languages and cultures. Its likely origin is that the thumb is often used for rough measurement by carpenters, seamstresses, artists and many others. In fact, the measurement of an inch is believed to have been derived from the distance between the tip of the thumb and the first joint. Rules of thumb such as the right hand rule in electrodynamics are also used as mnemonic devices. This usage, of course, is of more recent vintage.

There is no 'rule of thumb' for 'rules of thumb'.
As long as a piece of string.

Not really as it would depend very much on what you were taking a sample of.

For example, when canvassing opinions from people a sample size of 1000 is generally regarded as the minimum required for an accurate answer.

However, if you were sampling water from a tap for analysis you would only need to take the one sample.

In general, the more variables there are the larger the sample size should be but there's no hard and fast rule and often it's not possible to give an accurate figure as to the number of variables.