How can I insulate a basement wall that has 1x1 framing installed on it?
There is a rigid polyisocyanurate foam core insulation, often sold under the brand name Super Tuff from Dow that has a higher R value than XPS. It is R-13 in 2 inch panels and is higher if you trap a small air barrier behind it (around R-15).



The foam has a foil vapor barrier on the back.

You could 


mount this on the existing furring foil side out, using glue and screws
seal the edges with foil tape
reinstall paneling with glue and screws throught the foam, into the existing furring.


HOWEVER: The paneling could hold itself up when mounted this way, and you could mount lighweight things, such as pictures. The walls would NOT be suitable for mounting shelves, fixtures, heavy mirrors, etc. You would need to take care and inform any subsequent owner that the walls were not intended for bearing any signifciant weight!