The firm says customers who pay monthly will get back 10% of their July subscription, the equivalent of three days' charges, applied to their September bill.
Pay-as-you-go customers will get back 10% on their first top-up in September.
O2 said it recognised the failure had caused "inconvenience and frustration".
"We have now identified all those customers directly affected (those whose devices could not connect on our system) and we are giving them the equivalent of three days back for the disruption as a gesture of goodwill and to say sorry," the company said.
All customers, even those who were not affected, will also be given a £10 voucher to spend in one of its stores.
O2 admitted that about one third of its customers were affected by the network collapse.
Source: BBC News