Higher Intermediate Point.

Basically an enroute destination where the fares originating from there is higher than the original departure point.

ie fares from BKK to USA are cheaper than fares from HKG to USA travelling on CX. Thus HKG is a HIP in fare calculations, and a stopover of more than 24hours is not permitted unless you pay the higher fare .. as though you are originating from HKG.

On economy excursion fares, usually you cannot stop in HKG outbound but certain fares permit stops on the way back. This has nothing to do with HIP but rather exceptions to the rules set by each airline sales department.