The case
Cyndi Lauper's 1983 breakout was an MTV fixture, and the chorus is unmistakably "girls just wanna have fun." The title on the 45, on the "She's So Unusual" tracklist and on the Billboard chart is the formal "Want to."
Robert Hazard wrote and recorded it first in 1979 under the same formal title. Lauper's delivery turned it into the contraction everyone remembers, which is why the record-store clerk and the radio DJ never agreed on the spelling.
Why our brains do this
We remember songs by how they're sung, not how they're printed. The sung version becomes the title in memory.
Sources: Wikipedia — Girls Just Want to Have Fun · Wikipedia — False memory: the Mandela effect