You cannot dye a well-finished wood board. The oil will prevent dye from reaching into the wood.
If you sand off the finish, you can use food-grade dyes to color the wood. Since food dyes will not bind directly to cellulose, which is the material that both wood and cotton are made of, you will then have to seal the wood so that water cannot reach the dyes and cause them to run, blur, and wash out.
The only dyes which you can be sure are safe to use on food preparation surfaces are certified food colorings. Even non-toxic paints are not suitable for use on food preparation surfaces. "Non-toxic" on the label tells you that accidental ingestion will not cause death, but it does not imply that the paint is safe to eat.
The easiest form of food coloring to use would be markers that contained certified food dye. One brand that you can buy is called Food Doodlers. Use cautiously at first, dabbing the marker only lightly onto the sanded wood, to make sure that the color does not spread farther than you want it to. Test on scrap wood first.
A type of dye called solvent dye could be used on the oiled finish without sanding, but it is very definitely not safe to use on food serving surfaces.
An alternative would be to use a wood burning device. Practice a lot on less important pieces of wood before you even consider marking your prized heirloom.