Why did Boeing choose to give the Boeing 720 such a designation?
Because the launch customer, United Airlines wanted it. From Boeing frontiers:


  The 720 was a ... first marketed to the airlines as the model 707-020. United Airlines was very interested in the 707-020 but had previously decided to go with Douglas and the DC-8. To help United avoid any negative public relations for going back to the 707, Boeing changed the name of the 707-020 to the 720. 


According to aviationweek, the name change was at the behest of William ‘Pat’ Patterson, the president of launch customer United Airlines.