I'm sure some crazy modder could do it. However, it would be a very complex project.
You would also have to account for the space needed for the power supply you would need to power the LCD and the CCFL backlighting. Also, the AirPort and Bluetooth antenna on the Mac mini is the top surface. Putting it inside the dome would affect the range. And the optical drive would not be accessible inside the dome, unless you cut an unattractive slot; the optical drive in the iMac G4 is tray-loading, and uses a door opens and closes.
You'd also have to route the power connection and ports from the Mac mini to the dome somehow, or again have some unattractive holes in the dome. And the Mac mini may have cooling issues, if airflow was restricted inside the dome.
So I don't think it would be worth the effort. Better to preserve the iMac G4 for what it is, a memorable iMac design. You can run it normally and use it as a second display, with this clever program
http://www.screenrecycler.com/home.html
Then, you can have a real display as your primary and extend your Desktop to the iMac's screen.