I'd go for #2 and follow the directions on the Passport Office Web site. Unfortunately, it must be done in person at a Post Office or Consulate. They like details of the lost one if available (#, where/when issued) but I don't think that's a dealbreaker if you don't have them. "Renewing", to them, means you're just sending in your about-to-expire one and getting a new one. It can usually be done by mail. Not what's happening here. It's not a new application since she had one before (and the Passport Office may check that).