Can I sue the man that hit my car with my 6 year old daughter inside and the driver was arrested for DUI charges?
If you had any damages as a result of the hitting of your car (property damage or personal injury damage), then yes, you can sue the man. The basic elements of any tort claim are (1) an action, (2) that causes, (3) injury to another. As long as you have all three, you're good to go. However, before you do so, review your policy for exclusions. Some insurance policies will specifically exclude coverage for criminal acts (not all, but some). The reason this is important is that you may need your prosecutor's office to change the charge or sentence in order for you to collect from an insurance provider (either your or his). If you are not sure if this is a clause in your policy or not, arrange for an initial consultation with a personal injury attorney to look over your policy. From there, you can make the final decision of when and how to file your claim against the driver.
Regardless of which avenue you take above, also keep the prosecuting attorney's office updated on your out of pocket expenses. Some offices will actually try to collect restitution for you. Of course some are better than others, but if yours is willing and able to help it won't cost you anything for them to seek restitution on your behalf.