Top seed Agnieszka Radwanska led 6-4 2-0 in her last-eight tie with Dominika Cibulkova when the poor weather hit.
There was a resumption at 18:40 BST but the rain returned an hour later with the score at 6-4 3-2 before event organisers suspended play for the day.
Earlier, Monica Puig beat Kristina Mladenovic to make the semi-finals.
The Puerto Rican won 7-6 (8-6) 4-6 6-3 and will play either Radwanska or Cibulkova in the last four.
The other quarter-final match is between Elena Vesnina and Karolina Pliskova, with the winner to play Konta or Makarova.
That match was also affected by rain on Thursday, with Pliskova racing into a 4-0 lead before the players were forced off.
Persistent rain has brought play to a close at the WTA event in Eastbourne, where Johanna Konta is set to play Ekaterina Makarova in the last eight.