There are a couple of places you should check out for great music. These three often feature music that has ukes, banjos, mandolins... all that good stuff. Lot's of folk, bluegrass, old timey, old time country, americana, etc. Of course it varies so check out the websites for the show schedules to see if there's something you like.
manhattan:
Banjo Jims in Alphabet City:
http://www.banjojims.com/
National Underground in the lower east side:
http://www.thenationalunderground.com/ *best burger in the city in my opinion!... only comes two ways - all or nothing ;o)
Brooklyn:
Jalopy Theatre in Red Hook:
http://www.jalopy.biz/ This is a bit of a hike but it's defintely do-able by subwway and a little bit of a walk. or if you want to spring for a cab - you can get there door to door. This place is amazing! Absolutely love it and recently took a 6 week banjo class there. Combination vintage instruments, music school, and music venue. Shows every night except for monday. a great community and a lot of fun! and if your daughter isn't of bar age - i think that would be fine here but maybe call to check.
here's a video they made to show how to get there by walking from the subway
http://www.youtube.com/user/JalopyBrooklyn#p/a/u/1/-VXOVysAHc0
Feel free to hit me up if you have any questions!
Good luck, safe trip, and have fun!