It can be done!! You have to do this from one of the wireless handsets, not from the base set (makes no sense but OK).
Press the "envelope" button that says mute underneath it at the bottom of the handset keypad. It will start playing your messages. Press 5 to skip/pause this, then press 8 and wait for instructions to record. I got put off a couple times trying it at first because I didn't realize you have to listen to any saved messages you've got before it gives you prompts to change the greeting (or press 5 at any time during the messages). Then press 8 again to stop recording--it will play back your message. Press 8 to re-record or the END key to save it. That's it! Below are the instructions from the manual as well:
Your greeting can be between 2 seconds and 30 seconds
long. If you don't record an outgoing message, the system
uses a pre-recorded greeting: Hello, no one is available to
take your call. Please leave a message after the tone.
Press Messages/mute to access the answering system. Wait
until the system finishes its announcements, or press 5
(stop) to skip them.
Press 8.
Wait for the system to say "Record greeting," then begin
speaking.
When you're finished, press 8 to stop recording. The
system plays back your new greeting.
To keep this greeting, press End. To re-record it, go back to
step 2.