How do you find the name of a ship someone served on in WW2?

Assuming the person served in the US Navy, you should request the person's service records from the National Archives. The process varies somewhat depending on your relationship to the person.

http://www.archives.gov/veterans/military-service-records/#evetrecs

The records center receives nearly 5,000 requests for records per day, so allow a couple weeks for a response!