From Wikipedia:
'Til Tuesday frontwoman Aimee Mann briefly sings in each chorus of the song, marking Rush's first collaboration with another vocalist, with Rush guitarist Alex Lifeson adding that the band thought a female singer "would suit the song". Initially, the band had hoped to recruit Cyndi Lauper to sing the part, then later approached the Pretenders' Chrissie Hynde because, according to Lifeson, "we thought she'd be perfect. But Chrissie was unavailable at the time..." Rush later recruited Mann and paid her $2,000 to sing on the track, with Lifeson saying that "her voice blends with Geddy [Lee]'s perfectly and I think it creates the right atmosphere for the song. It's just something new for Rush..."