The phrase implies a gentle, but firm way to get someone to agree to do something. She was able to prevail upon her brother to attend the reunion.
Pro Tip 2/3
Persistence
Using 'prevail upon' suggests repeated efforts rather than a one-time request. He had to prevail upon them several times to get the signatures needed.
Pro Tip 3/3
Formal Context
It's a formal expression often found in written or official contexts, less common in casual speech. The council was prevailed upon to reconsider the zoning application.