Remove the pieces of the damaged flagstone with a stone chisel. Make sure to chisel toward the damaged flagstone so you do not damage surrounding stones.
If the stone cannot be removed with a chisel, break up the damaged stone with a hammer. Gently tap the flagstone until the pieces are small enough to remove by hand.
Remove the dried mortar pieces underneath the flagstone with the chisel.
Spread a layer of mortar in the hole and fit the replacement flagstone in the hole. If the replacement is too wide, measure the excess stone and use the hammer and chisel to shape the flagstone to the right size.
Apply mortar on the new flagstone to secure it in place. Use a trowel to smooth out the excess mortar from the sides. Allow the mortar to dry and harden overnight.