What is the name for a flight crew that includes extra members for a longer flight?
A crew that consists of the basic required cockpit crew for the aircraft plus relief crew for long flights is called an "augmented flight crew." As far as I know, this designation is used by ICAO, the FAA, and the EASA (and likely others.)

This ICAO document defines augmented flight crew as:


  Augmented flight crew. A flight crew that comprises more than the minimum number required to operate the aeroplane and in which each flight crew member can leave his or her assigned post and be replaced by another appropriately qualified flight crew member for the purpose of in-flight rest.


The FAA's Federal Aviation Regulations define it nearly identically in 14 CFR 117.3:


  Augmented flightcrew means a flightcrew that has more than the minimum number of flightcrew members required by the airplane type certificate to operate the aircraft to allow a flightcrew member to be replaced by another qualified flightcrew member for in-flight rest.