I think you might just be out of luck on this one. But not necessarily!
The magnetic tape you talk about sounds as if it's made up of the
same kind of stuff that makes up refrigerator magnets (it has to be
flexible, after all, and have a magnetic field strong enough and
varying in space enough to make stuff stick to it magnetically). It
probably has alternating N and S poles along its length (try putting
some iron filings on a strip of it to see where the field lines point).
Anyhow, if the roll of magnetic tape has been coiled up long
enough, it can demagnetize itself, especially if repelling poles are
forced to sit side by side for long periods of time. The tape was
probably magnetized in the first place by magnetizing hot iron filings,
cooled down past their Curie temperature (very hot!) in a magnetic
field. Then these filings were mixed in with molten plastic or rubber,
and this was cooled down in a field with alternating N and S poles
along the length of the tape. The filings orient themselves along the
applied magnetic field, and then get stuck in the cooled plastic. Over
time, they will demagnetize. The best way to get them to remagnetize is
to repeat the process that got them that way in the first place, but
I'm afraid you'll make a sticky mess as all the gooey plastic melts,
and you may burn it all before getting up to the Curie point of the
Nonetheless, you could try putting it in an externally applied
field, to see if that helps. I suspect it'll be tough going to get a
good strength out of this, especially if you are reversing the
direction of the existing residual magnetization.
I'd buy another roll of the stuff. It's gotta be easier that way. Probably cheaper too.
P.S. half-inch magnetic tape (not the sticky kind) is what lots of
physicists used to record the data from their experiments! We still use
wide magnetic tapes, but they are now wound up on spools inside plastic
(published on 10/22/2007)