Can a person sue a designated driver if the DD took them to their car and not to their home and then they got a DUI?

In your fact pattern, you stated taht your daughter had a "designated driver" at the time she went to a party.  After the party, the designated driver drove your daughter back to her car, after which, she drove the vehicle and hit a tree.  She was subsequently arrested for DUI.  Your question is whether she can sue the designated driver.  The first question to be answered is whether there was any legal obligation on the part of the designated driver to drive your daughter home.  Apparently, your daughter drove her car from her home to another location from where she was able to ride with the designated driver.  This may be an indication that your daughter was planning to drive her car home from the location where she parked her vehicle.   Accordingly, unless there was some binding  contractual obligation for the designated driver to drive your daughter home, then chances of prevailing in a lawsuit are questionable.  Second, as with any other driver who is arrested for DUI, it is the driver who makes the decision whether or not to drive after drinking.  Without more facts, you should consult with an attorney to provide missing informantion.