The entire incident was captured by on-board video that was posted on YouTube.
According to the Los Angeles Times, the boy, identified as Kyle Wheeler, underwent multiple surgeries as well as physical therapy before he could walk again. He has since recovered, the family's lawyer, Art Morgan, said Friday.
Morgan said the settlement covers the cost of medical care and the family's emotional distress.
A state investigation concluded the ride hadn't been properly maintained.
Shortly following the incident, the Buena Park amusement park said in a statement that the cable on the Xcelerator ride is changed regularly. Park officials said the cable was last changed in December 2008 and wasn't due for another change until December 2009.
The Associated Press contributed to this report.