I think they need to do the following:
1 - Offer a genuinely humble apology.
2 - Go into broad details about what they've done to the PSN to make it more secure, so that the internet tech heads can confirm that it's top tier security, and that network security is being taken extremely seriously going forward.
3 - Confirm that the new PSN will allow for more advanced features, with details to be revealed at E3.
4 - Offer PS Plus users an extra 2 - 4 months without charge, and an extra game this month.
5 - Give all PSN users two free PSN games from a reasonable selection of titles.
When Sony announce that the PSN is back online, they will have their best chance to hit the headlines again, so they need to ensure that there's nothing but good news to report, with a lot of rock solid assurances made.
Acting quickly and decisively with remarkable generosity is going to be the best way to turn this around. Ideally they need to make the compensation so remarkable, that talk of the compensation quickly overshadows the condemnation they've drawn throughout the incident. It's the best way to get public opinion back on their side, but heaven help them if they just settle on handing out minis. Now is not the time to be cheap.