Hi all - I think I found two work-around ideas.
First, I appreciate the input on filenames. I re-ran it with the simplest names I could give it and got the same error. Here is the "copy as Python Snippet" for what I tried:
arcpy.gp.Filter_sa("C:/temp/avg.tif", "C:/temp/avg2.tif", "LOW", "DATA")
It feels like we can eliminate wacky file names from the list of suspects.
The error is saying the neighborhood mask is invalid, which I'm guessing is basically a kernel file, and if it's missing, that would be invalid... Why it's missing is anyone's guess, but the name looks like a temporary file name, so I started asking myself "what would cause a temporary file to fail?" Well, I'm not running ArcCatalog as an administrator user, and that would absolutely prevent me from writing into the program files x(86) folder. In fact, I had that problem the other day with something else.
WORK-AROUND 1: Run as Administrator - This worked!!! Except, it's a real hassle, because it actually runs as another user, so I don't have access to any of my settings (database connections, folder connections, symbols, etc, etc.). I guess I could switch over to the administrator user full-time, but... yuck.
WORK-AROUND 2: Can someone please confirm the following?
Is the FILTER tool with the LOW type the same as FOCAL STATISTICS with the RECTANGLE neighborhood at 3x3 and a MEAN statistics type? That is working fine for me.
Last note: I appreciate the suggestion to file a tech support request, but our academic site license policy is basically that we are only allowed to do so when it is a total show-stopper, so I'm wondering if FOCAL STATISTICS is the same thing.