WHat is the best method for applying lemon oil to a fretboard ?

I’m not sure there’s a wrong way (short of just pouring it over the board) but I use a cotton bud and kind of “paint” it on a drop or two at a time.
I let it sit for a little while (10-15 minutes, depending how quickly it soaks in) then wipe it off with a bit of clean cotton rag.
I’ll repeat if needed.

windmill, you are on the right track. 10 or 12 drops is plenty for an entire bare rosewood or ebony fingerboard. More is not necessarily better because probably 90% of the oil stays on the cloth or paper towel and it goes into the trash.

I put a few drops on the fingerboard and then wipe it on with a fingertip wrapped in a layer of cotton terrycloth until all the wood is coated. I follow this up by rotating to a clean/dry portion of cloth and continuing, rotating to a clean portion of the cloth as it gets funky. Keep scrubbing with a clean section of cloth until the wood is completely dry; don't leave it wet with oil.

It's not good practice to wipe the oil on and let it soak in before wiping it dry. Don't allow it to marinate. This is especially important if the rosewood/ebony board has a finish along the edge where the fret ends are. If you allow the oil to soak in, it can cause the lacquer to lose its grip and the finish could flake off.

When oiling the fingerboard on an acoustic guitar it's good practice to treat the bridge if it's unfinished. The only caution here is on older guitars: do not allow the oil to puddle along the joint where the bridge meets the top. Oil can work its way in and loosen the glue bond. To prevent this, I apply a drop or two of oil directly to the cloth and then scrub away.

You can do the same with an unfinished bridge saddle/foot on an arch-top guitar. If the guitar has an unfinished wood trapeze tailpiece, hit that too.

I'll stand corrected. Though I've never experienced any of those adverse effects mentioned.
But I also wouldn't say I "marinate" the board either. Often times using my application method, by the time get 4 or 5 frets up with the cotton bud (Q-tip for you in US) the oil on the 1st & 2nd frets has lost its sheen. It's not like it puddled or pooled on there.

And like wrnchbndr, conditioning is always done after finger funk removal with naptha.

I put a drop on each board section between the frets and then rub it in.

I would be very conservative with it. I messed up a perfectly good Les Paul using lemon oil. The location up high on the fretboard where it joins the neck raised up absorbing that stuff and it took some work by a local guy to fix it. I don't do it anymore. just use the Dunlop fret cleaner now.

I've used 3-n-1 oil. applied with a cotton swab, against the light to ensure the minimum but even coverage, then buff off with paper towels. Worked a treat.

The rule of thumb for all liquid oils is "minimum needed and no more".

The same applies to tung oil, neatsfoot oil, Danish oil, etc. Most are not what they purport to be.

For many years I used Hawaiian Tropic Dark Tanning Oil.  It's mostly mineral oil, and my guitars smelled like the beach.

A really good alternative to lemon oil is linseed oil. Boiled linseed oil air dries and hardens. It's the traditional finish for fine stocks on rifles and shotguns because it's weatherproof and easily repairable. Unboiled linseed oil does not harden, and it's probably better for a guitar fingerboard than the boiled type. You can find this stuff inexpensively in the paint department at most any hardware store. If you buy it in an art supply shop, it's usually twice the cost.

I snagged some of that oil, as well, and it’s been great. I’ve been using it at least 2 years now. Maybe 3.

I recently refurbed my Carvin Strat and added a couple coats of Howard Feed-n-Wax to the tung oil finish while I had it apart. I decided to try it on the ebony fretboard. I used it sparingly and then polished it dry. I really like the results. The fretboard is very fast but it doesn't affect the strings like most oils. I may try it on some other guitars.