I like Miller's Pub on Wabash. Nothing in the way of deep dish etc. but a solid, old school Chicago restaurant with a nice atmosphere and decent fare. They've been around 75 years.
(I like the bar too.)
You can order food until 1 or 2 in the morning and/or drink until 4 am if you are so inclined.
134 S Wabash Ave, Chicago, IL 60603
Yup, Pizano's is THE place for deep-dish near the Palmer House. (And even us longest of long-time Chicago residents can sympathize with occasionally saying Wacker when we mean Wabash, and vice versa!) Phone ahead with your pizza order to avoid waiting 30-45 minutes while seated for the pizza to bake. www.pizanoschicago.com
Here's one more place nearby for "signature Chicago cuisine". Garrett's Popcorn is a local specialty, for their freshly-made caramel popcorn (with or without cashews or pecans), cheese popcorn, or a mixture of the two. The Palmer House Hilton is right between the location at Madison and State (a block north of the hotel) and the one at Jackson and State (a block south of the hotel). www.garrettpopcorn.com
There are a lot more great places to eat near the hotel, although they are not necessarily foods unique to Chicago. For breakfast recommendations near the Palmer House Hilton, see:
For lunch recommendations near the Palmer House Hilton, see:
For cheap eats recommendations near the Palmer House Hilton, see:
For late night recommendations near the Palmer House Hilton, see:
For recommendations of new restaurants near the Palmer House Hilton, see:
And for more on foods that are unique to Chicago, see: