Spray-paint a box that measures 2 square feet onto the floor in the lowest corner of the basement.
Cut through the concrete inside the spray-painted lines. Use a jackhammer to remove the concrete until you get about 6 inches away from the walls, then use a hammer and chisel to remove the rest. This prevents you from damaging the walls with the jackhammer. Wear safety goggles and ear protection when you use the jackhammer.
Remove the chunks of broken concrete, then dig a hole in the soil beneath the floor. The hole should be at least 8 inches away from the wall, so that you don't damage the footings that support the slab. Dig the hole 6 inches wider than the sump pit liner -- and as deep as the liner, or greater. The liner is a plastic container covered with holes that allow water to flow into the pit.
Place the liner into the center of the hole, then pour gravel into the space between the outside of the liner and the soil, completely filling it. The gravel acts as a filter to prevent dirt from entering the liner and clogging it.
Pour gravel on top of the soil around the hole until it's an inch higher than the bottom edge of the slab, smoothing it when you're done.
Place plywood boards next to the wall around the outer edge of the square you dug in the floor, if you have a floating basement floor. (If there's a trench between the slab and the wall, you have a floating floor.) The plywood will create a trench when you cement the floor. Spread motor oil on the bottom of the plywood along the inner edge to prevent the concrete from sticking to the wood.
Mix concrete, following the manufacturer's instructions. Pour the concrete on top of the gravel until it's even with the surrounding slab. Smooth the surface with a trowel, then allow the concrete to cure for at least two days. Remove the plywood after the concrete has cured.