How can I extend a cable to move an outlet inside a kitchen island cabinet?
First of all, figure out if the original wire is stapled or not.  If it isn't then the solution is easy.

CAUTION: Requires work IN the panel.  If you are not 100% confident you can do this safely, call an electrician.   It's not difficult, but you need to know exactly what you're doing.


Turn off breaker.
Cut pretty close to the box.
Use this slack to do what you need to do upstairs.  -- free fixed outlet, new wire to the one you want to move. 
Mount a junction box a couple of feet from the panel in the basement.
Find a piece of cable that will reach from the new junction box to the panel.  (Maybe there's some old wire we just removed lying around?)
Splice in the junction box, and find someone to connect it back to the panel.