Is this a custom geoprocessing service that you created? To improve/modify error handling and their associated error messages, you may be able to modify the Python script that gets executed by the GP service.
Sometimes Esri error messages appear when a tool simply fails; the resulting error message is something that is embedded into the Esri code for that tool and it can't be changed. I think most of the time, we're all at the mercy of the Esri error reportnig that is part of the application code. We've all seen the dreaded "General System Failure" or "99999" error message from Esri tools that haunt us from day to day.
Other times, if there is a failure in your own script somewhere, you may be able to provide a clearer error message by inserting the arcpy.AddWarning or arcpy.AddError module throughout the code. Depending on the version of ArcGIS you are using, these links may be helpful for you:
http://help.arcgis.com/en%20/arcgisdesktop/10.0/help/index.html#/AddError/000v00000003000000/
http://resources.arcgis.com/en/help/main/10.1/index.html#/AddError/018v0000005w000000/
http://resources.arcgis.com/en/help/main/10.2/index.html#/AddError/018v0000005w000000/
Think of these as custom error reporting where you, as the script programmer, anticipate which issues may occur based on script input and then modify your code to better communicate those issues to the end user.
I'm not sure if this answers your question, but I understand that error reporting can be frustrating.