Measurement of groundwater levels
Terminology

The term "average" isn't a specific term.  Are you talking daily average or annual average?  The question "What is the average?" begs the question of "Across what timeframe?"

To explain that point, let's look at a chart:

 

This is the groundwater level of a well in Georgia (US).  The "Average" for this well in December 2003 would be 176.9 feet below ground level.  However, for the year 2003 altogether, it would be around 177 feet below ground level.  For the previous decade, the average would be around 175 below ground level. 

Timeframe

To answer the question, the best timeframe is the largest timeframe you can afford to gather.

If we look at the Raw data for the above chart, what we see is that for the year 2003, the groundwater levels went up!  If you only gather one year of data, this well would show a dramatic rise in water levels, which would clearly not be accurate.  

If you extended the search back five years, you would see a large downward trend.  March 1998 shows 173 feet depth and March 2003 shows 178 feet depth.  This trend indicates a dramatic loss of water volume.  However, without a larger timeframe, it's impossible to tell if this is just a negative trend or a cyclical one.

Even in the chart above, the mid-1960s may just have been a high point in the water table.  (Indeed, Georgia saw 15% more percipitation in 1967 than is historically average and 1999-2001 were historically low.)

Summary

The best timeframe to gather data is the longest timeframe that you can get.