My spouse and I purchased a home 15 years ago at the time ( in Georgia
)  , a friend
also was named as a purchaser and on the deed - for purposes of
qualifying for the loan ( the friend was a resident of the state at
time of purchase - we were not - we became residents in 1 month ) .
After a dispute several months later the friend left the house and
ultimately the state and was not heard of for 2 years , at which time
they requested that papers be signed releasing them from financial
obligations on the mortgage - this was done . 10 years later while
attempting to secure a refinance on the property we found that the
person was on the deed for the home . When asked to sign a quitclaim ,
they refused we let the matter drop and secured funding elsewhere  .
Now we are again trying to refinance - what are our options to
regaining control of the property?

If you have a copy of the document which released your friend from
obligation, that might surfice. Since your friend was at substantial
financial risk for 2.5 years (and minor risk for 15 years) it would be
fair for you to offer him $1000 for the quit claim deed you want in my
opinion.   Neil