Granite makes an excellent choice as a floor tile because it's one of the hardest stones available. Other types of stone, such as marble and soapstone, will not cause damage to granite but pieces of glass dragged in with foot traffic might. Performing a surface repair by polishing the stone is a difficult job that should be undertaken by a professional as an improper technique can cause more damage than that which already exists. For small scratches, however, you can try repairing it yourself.
Check that the mark is actually a scratch rather than an imperfection in the stone. Look at the scratched floor tile from different angles in the sunlight to see how deep the scratch is. If there are a number of marks and imperfections, it may be time to get a professional to seal and polish the tile for you. Don't try to do the polishing yourself as you may cause a dent to form in the tile or the surface and this could lead to it becoming permanently damaged.
Spray some non-rinsing stone cleaner onto the tile and leave it to sit for about 5 minutes. Use a microfiber dust pad to scrub it clean, then dry it off.
Use a dry steel wool pad in a circular motion to buff the surface of the tile. Keep the pressure light and extend the motion to take in the area surrounding the scratch as well. After, clean the area again and allow the granite to dry completely before using a soft cloth to polish the surface.
Fill Deep Scratches
If there are any deeper scratches on the surface that have not been eradicated by buffing with the steel wool, you can fill them in using grout mix. Mix the grout according to the manufacturer's instructions and apply it to the tile. Press the grout into the tile and smooth it off so it is level. Leave the grout to dry for 24 hours, making sure that nothing is laid on top of it. Apply two coats of grout sealer 48 hours after application to protect the surface from future damage. Allow the first coat to dry fully before you apply the second.
Paint Discolored Areas
For any areas that are still a slightly different color, use a paint marker or permanent marker to color the marks in carefully. Pat the area with a cloth to remove any excess ink then clean the whole floor with granite cleaner and leave it to dry overnight. Inspect the scratched area in the morning and make any additional repairs with the marker.
Use a water-based sealant to reseal the whole floor to ensure that the finish is uniform and the damaged area is completely protected from future damage. To help prevent future scratches, use doormats, or remove your shoes upon entering the house.