There is not currently a way to log the telnet/ssh output other than copying it with the copy command at the bottom of the window.
Personally this is why I use a (free) external tool to cover my Telnet/SSH needs.
If you use puTTY it is possible to change the default settings such that it will log the output of a session to a text file.
When you start puTTY just click on the "Logging" option under "Settings", I tend to choose the "All session output" option, then choose a file location (I tend to use the given % values to make filenames unique for device and time). Go back to the main Session page, click "Default Settings" then click "Save" so the default action is to save all session output.
You can then create a new "External Tool" in SolarWinds using the path to the puTTY executable and, in my case, click on "IP Address" for the parameters to send. You should then find if you right click on a device and go to the "External Tools" part of the context menu your entry for puTTY is now there, and when you click on it you get your telnet session (or whatever your default it.)
Nice tip :)