Since St. Patrick's Day is on a Sunday, you have the choice of all of the many Sunday brunches around town. Most Sunday brunches tend to be busy starting around 9:30 at the breakfast-focused restaurants (most of which don't take reservations or serve alcohol) and around 10:30 at the dinner-focused restaurants (which usually offer reservations and alcohol), then stay busy till around 12:30-1:00. I don't know which might have music.
You'll find lots of detailed reports for many of those brunches in the brunch topic at www.chow.com/topics/364403
If you'd like some specific suggestions, let us know what you're looking for. Some of the ways in which you might narrow things down are location (downtown? neighborhood? suburb?), type of food, price range, etc. Some of the places I've recommended in recent discussions for someone looking for reservations - which means the places most likely to serve bloody marys - included North Pond, Perennial Virant, Cafe Ba-Ba-Reeba, Deleece on Southport, Big Jones, and Nightwood. I would also add Shaw's Crab House, with their all-you-can-eat buffet including outstanding thick-cut caramelized bacon, Alaskan king crab and shrimp cocktails, crab cakes, creme brulee, and chocolate pots de creme; they also feature a bloody mary bar. It was $48/pp (excluding alcohol) last time I went.