I am not sure how a 4th of July weekend is cheaper than all the rest, but that may be the best option there for price alone. However, if hotels in Chicago are anything like ours here around 4th of July, they are super busy! Due to this fact alone, doing it after the 4th of July weekend may be a better option.
Personally, I think any are fine, but cheaper is always better right, but be prepared to get a few canceled reservation due to unforseen "4th of July" events in the family.