what engineering properties do you need? once you settle on that, than pick a material. and generally it will be a function of cost effective local borrow materials
Check your state or federal regulations the cap of a landfill has to provide a low permeability barrier, which is why most are being capped with compacted clay or HDPE liners. Landfill gas is also captured in larger facilities and needs to be considered in cap design.
yes, there are requirements for final cover which generally includes an infiltration resistant layer (with permeability requirements) and a top layer of erosion resistant material. thickness of the infiltration layer will be related to the permeability. Usually these materials are stockpiled from the onsite excavation for the landfill but if not they need to be borrowed and hauled in from somewhere else.