One team that jumps to mind is the 1996 Florida Panthers. A third-year expansion team, the Panthers made their playoff debut that spring and shocked Boston. The underdog Cats then upset Philadelphia and Pittsburgh to advance to the Stanley Cup Finals. While the Panthers were swept by Colorado (in their first year in Denver after moving south from Quebec City), they remain one of the most popular teams in South Florida sports history.
I think a big reason that team remains so popular is because how they did things. No one expected the Panthers to do anything when that postseason started, and they just took everyone around them for a ride. It was good stuff. But the Panthers didn't win. What great underdog team actually took home the Cup? Carolina and Tampa Bay could be lumped in here, but those were pretty good teams no one outside of Tampa or Raleigh took seriously because they were from Tampa and Raleigh.
Don't think you can lump either Ottawa nor Anaheim in the underdog category. Anaheim was the trendy pick to advance to the Finals from Day 1, while Ottawa's been knocking on the door for years and had a terrific second half. Should be a great Finals.