The company said its 3G and 4G data networks, which were taken offline by a software issue on Thursday morning, were restored to normal on Friday.
Some O2 customers also reported being unable to send and receive text messages but O2 said this issue had also been resolved.
The firm said it would be giving its pay monthly, small and medium business and mobile broadband customers credit for two days of monthly airtime subscription charges, to be issued by the end of January.
Pay-as-you-go customers are to be given 10% credit on a top-up in the new year, O2 said, adding it will let customers know when this is available.
O2 said pay-as-you-go mobile broadband customers would receive a 10% discount on a Bolt On purchase – which are additional call, text or data bundles that can be purchased at any time.
“We’re very sorry about yesterday’s data issue. We understand how important it is to stay connected, especially at this time of year,” a company spokeswoman said.
“We’d once again like to thank our customers for their patience. We’re doing all we can to make sure this issue doesn’t happen again.”
Sky Mobile – which uses O2 services – has announced plans to give its customers a day’s free unlimited data on Saturday December 8.
On Thursday, O2 issued a joint apology with telecoms company Ericsson over the mobile data incident.
O2 UK chief executive Mark Evans said: “I want to let our customers know how sorry I am for the impact our network data issue has had on them.
“We fully appreciate it’s been a poor experience and we are really sorry.”
Marielle Lindgren, chief executive of Ericsson UK and Ireland, said: “The faulty software that has caused these issues is being decommissioned.”
She added: “Ericsson sincerely apologises to customers for the inconvenience caused.”
Alex Neill, Which? Managing Director of Home Products and Services, added:
“It’s right that O2 compensates its customers for the frustrating network failure suffered by millions of customers. Anyone who suffered out of pocket expenses should make a claim to their mobile provider.
“In addition, O2 needs to give its customers reassurances that it is taking measures to stop this from happening again. Connectivity is now such an integral part of our lives, it is time for the regulator to consider whether it should introduce automatic compensation for the inconvenience caused by severe outages.”