China's Jie Zheng was leading Roberta Vinci 6-3 5-6 when rain interrupted their match and forced the players off the court.
Zheng took the opening set 6-3, but Vinci responded impressively in the second set before rain intervened.
But Misaki Doi did manage to wrap up victory before the downpour ended play as she secured a 6-4 7-5 win over Stephanie Foretz Gacon of France in the third round.
Doi will next take on Jelena Jankovic in the quarter-finals, on the same side of the draw as Vinci and Zheng.
Organisers at Edgbaston will be anxious for an improved forecast on Saturday.