What can I use to shore up ground eroding under a concrete patio? Eventually grass will be butting up against patio

I would guess that the most effective (and probably the most economical) remedy is to dig a perimeter footing at, and at least partially, under the edge where the erosion is occurring and pouring a new concrete footing/foundation (maybe 12"-18" deep and 6" wide) that will hold the underlying soil base from moving out from under the slab.

You Can Try Using Sand Rock Or Crush And Run Stone These Can Compact Well