Break the mortar that surrounds the damaged flagstone using a stone chisel. When chipping away at the mortar, point the chisel in the direction of the damaged stone to avoid damaging any neighboring stones.
Remove the broken up mortar, and break up the damaged stone with a hammer. The stone will be easier to remove after it has been broken into smaller pieces. Hammer lightly at the stone so as not to disturb nearby flagstones.
Remove the stone, and chisel away at any existing mortar that was surrounding the damaged stone or mortar that is in the bottom of the hole.
Fit the replacement stone into the hole where the damaged stone was located. If the replacement stone is too large, use a chisel and a hammer to form the stone into the approximate shape of the old stone. Make sure that the stone will be large enough to fit inside the hole on top of the mortar. If the stone is too small, it will not be level with the rest of the stones.
Use a shop vacuum to remove any dust or rock particles in the bottom of the hole for the flagstone. The bottom of the hole should be fairly clean in order to pour new mortar.
Dampen the inside of the hole by spraying a small amount of water. Mix some mortar, and spread it along the bottom of the hole. Firmly press the new flagstone into the mortar. Place a straightedge over the new flagstone to make sure it is level with the neighboring stones. If the stone is level, let the mortar dry over the next 24 hours.
Use another batch of mortar to fill in the area around the new piece of flagstone. Dampen the area around the flagstone. Fill the joints with mortar by using a grout bag. Smooth out the mortar that has been added to the joints using a small trowel. Let the mortar dry overnight.