Is it possible to use the general purpose mouse in a terminal
or console to copy and paste to a file?

Copy/paste with gpm seems only to work to paste in the terminal.

I would like to be able to copy terminal output, such as an error
message (one tha does not make it to any log), and paste it into
a file or some other permanent retention.

Yes, you can copy/paste into a fie using gpm. Highlight the text you want using the left mouse button and then open an editor and paste, using wither the middle button (or wheel) or both buttons (if it's a 2 button mouse) and save the file.
But you cal always redirect logs or the output of a program into a file using the ">"

Which editor? I tried Midnight Commander, but that only has copy and move commands
for the current file loaded into its editor. I also cannot use any of the editors
in KDE: seems these use their own separate clipboard and paste something other than
what gpm stored, which I guess is not surprising.

Again which editor: vi, joe, emacs?

How might I use ">" or "<" to redirect the right mouse button output in the console?

You can use vi (in edit mode if you want to paste), joe, nano (pico) etc.
If you want to redirect logs or the output of a program you can use ">" to put it in a file:

not sure what i would do is hold shift and left mouse button to highlight, then use nano -w /path/to/file once in the file (if it does not exist nano will create it) hit the middle mouse button to paste ctrl-x to save.

now if you use ctrl + alt F1/F2 etc.. to go to another console you will be reading from the same clipboard unless that console is running X then you will be reading from X/gnome/kde's clipboard etc.. also leaving any console will chop of anything on screen so you can no longer scroll up. but basicaly, if you want to copy and paste from a console not running X nano should work, it has no gui so X is not needed.
sorry if im talking about the wrong thing.