The question of which Hilton is sensible but still all three arevrather short cheap cab rides away so you can consider many options. Chicago is an easy city to navigate.
For "new and fun" I'd definitely suggest Sable which has a great service team, superb cocktails and comfortable bar for women and a booth facing the kitchen is great fun and wonderful food.
Another real favorite - not so new but neat room, good vibe and great food would be Mercat a la Planxa.
Mercat a la Planxa
638 South Michigan Avenue, Chicago, IL 60605
Sable Kitchen & Bar
505 North State Street, Chicago, IL 60654
I guess it depends on how "near" the OP is looking for. For example, Sable isn't within a 10-minute walk of any of the Hiltons; it's over half a mile from all of them. As much as I like Sable - it's my favorite contemporary American restaurant in the greater downtown area, and the artisanal cocktails are amazing and fun - there are other excellent places within a block or two of each of the Hilton hotels, that can be walked in about three minutes so there's no need for a cab ride. Your choice.
Near the Palmer House Hilton, I'd go to Vivere for creative Italian cuisine. www.vivere-chicago.com Atwood Cafe is also close by for contemporary American. www.atwoodcafe.com So is Rhapsody, also for Italian. www.rhapsodychicago.com
Near the Hilton Chicago, Mercat a la Planxa, mentioned above, is an excellent choice. It may be our best tapas restaurant. www.mercatchicago.com
Near the Hilton Suites, I'd go to Cafe Spiaggia for Italian cuisine. www.cafespiaggia.com (it shares the website with its expensive sister restaurant, Spiaggia; click on CAFE)