From the Dan Erlewine video, he is recommending Fretboard Finishing oil.
Going to the StewMac website, and looking at the description of Fretboard Finishing Oil, I have extracted the following paragraph (strictly quoted, no edits):
"Pure linseed and mineral oils have been traditional fingerboard preservatives, but they don't dry completely hard, often leaving a sticky playing surface. ColorTone Fretboard Finishing Oil combines the nourishing benefit of traditional linseed with special dryers. 24 hours after application the wood is sealed and protected. The fretboard looks great and feels smooth, never sticky."
So we have two premier guitar repair experts, Frank Ford (recommends mineral oil) and Dan Erlewine (recommends the StewMac Fretboard Finishing Oil, see other comments above). Two great sources, in disagreement.
Loads of opinions, most of them divergent.
The talk of furniture oil (Watco, Danish, and others) is off base. Reason it is off base is that furniture oil is not designed to have metal wires rubbed against it, like fretboards will have if the guitar is played, by definition. I don't know how it came to be that we needed some sort of coating or vanish to be on the surface of the fretboard, but between the steel strings, and the sometimes acidic hands of some players, the coating will turn to goo. Using an oil intended to sit in place and look pretty is not what is suitable for fingerboards if you look at the use case realistically. The furniture oils were not intended for this use, and are not well suited to it. You need to seal the fretboard to the greatest possible extent by getting a thin, drying oil into the pores of the wood, not sitting on top of it. That is why I made the recommendation for natural (non-polymerized) walnut oil. I don't want my fretboard to look like it is varnished; I want it to look like the wood that it is to the greatest possible extent. If I wanted a varnished fretboard, I would have it varnished. I like the feel of a bare wood fretboard, and will do what I can to prevent it from drying out. The idea of putting oil on a fretboard is not to coat it, it is to seal it against moisture loss.
As I have said before, if you want to coat your fretboard in axle grease, feel free. You own it, you play it, you need to deal with the consequences of your actions.