Lincoln Park for a family day, Bucktown+Wicker Park for hip shopping and fun, Gold Coast for the best of the best, Wrigleyville for group fun, and the South Loop for business travelers.
In addition to the above, neighborhoods like Lakeview, Lincoln Square, and Andersonville. In these neighborhoods, you will be further out from the center of the city, but you will have more of a taste for the local feel. Lakeview is great because it is close to the Lake Michigan. Lincoln Square has one of the cutest retail strips along Lincoln Ave for Airbnb foodies and shoppers. Andersonville is in close proximity to a number of colleges on the North Side.
Most of the North side neighborhoods are not convenient from O'Hare airport on the "L" because you have to go downtown and then back out again. To go to a North side neighborhood, like Lincoln Park, you are better off getting a cab, Uber, or taking the Blue line "L" to Western and then taking the bus due East.
One of the most up and coming neighborhoods is Logan Square. For business travelers, this area is a bonus because it is almost equal distance from O'Hare and Downtown.
For the neighborhood feel go with Old Town, Lincoln Park or Wicker Park (off Division Ave or Milwaukee Ave between Division and North). Those area have a lot of shops and restaurants to check out. A lot of good restaurants are in Logan Square.
Checking Compliance of Neighborhoods/Buildings
In addition to doing your research on which Chicago neighborhood the Airbnb is in, be sure the rental is city-compliant. Hosts have to be registered, and the building itself cannot be on the "prohibited" list (many condominium buildings do not allow short term rentals). This is important, since if the rental isn't valid, there is a chance you will be prevented from staying in it when you show up.
Airbnb is not welcome in most downtown Chicago high rises. You should stay with the neighborhoods outside of the Loop. Be sure to ask your potential Hosts and have them verify the compliance of their units. Better this, than getting kicked out mid-stay.