Andre Pearson, a heart failure patient who has been at Mayo Clinic since March, initially was too ill to leave the Rochester, Minn., hospital to go to his daughter’s wedding in California. He had resigned himself to watching it online. But a few days before the Saturday evening ceremony, with Mr. Pearson doing well, his care team decided to explore whether it might be possible for him to make the trip after all. With the right game plan in place, they determined that it was. A Mayo staff member accompanied him, and Mr. Pearson, an ordained pastor, surprised his daughter, Alexandra, by arriving the evening before the wedding in Indio, Calif., and promising to walk her down the aisle.