I have a series of records of annual maximum daily rainfall for a station for about 80 years (1935-2014). Only one value is very high compared with the others, but in this year, the other close rain-gauges didn't record such a high value. What are the conditions/tests to remove or keep this high value in the statistical analysis of rainfall data?
As you have multiple nearby gauge stations. I would suggest to compute a multiple linear regression that would allow you to estimate the rain in one station using the data from the others.
Then, compute the historical standard deviation of the measurements in that station relative to the prediction using the linear regression model.
With this, you can filter data errors by rejecting all points that deviate more than X standard deviations from the prediction based on nearby stations. You have to define your threshold X based on experience, and by looking at the data. It will depend in the quality of the correlation between the precipitation in the different stations, so no universal value can be defined.
The comments suggesting there are lots of ways to remove extreme values but also a great need for caution with precipitation is correct. I am not aware of any good way to test for this from a hydrological perspective, and ~58 mm/day is definitely possible, especially if a region has convective storms. Your best bet for verification is to check nearby gauges and historic storm records if they exist, although there is a very good chance that this is real data and therefore not desirable to remove (even if nearby gauges do not show a similar event that year).
If this a single gauge and you want to use it for modeling, you may wish to look at areal reduction factors to scale precipitation from a single gauge to a watershed scale instead of removing the data point.
It depends on what you want the info for. If you are designing a dam or drainage system for a 500 years event you better not ignore the high points. I recently measured 40 " in my yard during "Harvey". This is dramatically more than I measured in any other storm for 20 years but that doesn't mean it didn't happen.