Custom House, recommended above, is an excellent upscale venue, and not too far away. If you want to go REALLY upscale,
Everest has private rooms too, one of the top restaurants in the entire city and sure to impress.
Two nice places in the Loop with private dining rooms of sufficient size are
Trattoria No. 10 for Italian food, and
Atwood Cafe for American food.
At the opposite end of the spectrum, not upscale at all but they have a private room as well as great food, is
Lao Sze Chuan, for Chinese food in Chinatown. Another not-exactly-upscale option is one of the Greek restaurants in Greek Town (listed
here), most of which have private rooms.
By cab from McCormick Place, Lao Sze Chuan is under 5 minutes, Custom House and Everest are 5-10 minutes, and the places in Greek Town and the Loop are 10 minutes.