I'd wipe it thoroughly with lint free dry cloth to get all possible oil off without taking off black color, then give it a few thin coats of spray lacquer satin finish. That can protect it from rubbing off.
For the most trouble free fingerboard, I would go with stripping the black off and letting the natural wood show. Be a little different, have a brown fingerboard if you dare. :^)
For a more durable stain try minwax wood finish in ebony color. If you want the fingerboard very black, strip the oil dye off thoroughly, sand it clean and smooth after using stripping solvent, then it will be ready to stain. Don't sand above 320 grit as the wood may not accept the stain as well.
Takes several coats to get it black but it will get very black in 4 coats. Once it does, it dries hard and has not rubbed off on my hands yet. Follow directions closely to rub off excess within 15 minutes once wiped on. That goes for each coat. And let it dry well, 4-6 hours recommended, between each coat.
Tom