Before you start drilling, lay a strip of 2" wide masking tape over the area where the hole's going to go & make sure it's smoothed down & tight - to avoid splintering out any of the wood grain.
Also, use a second drill bit, that's a little larger (the same size as the O.D. of the swivel eye shank) and enlarge the top/outer 1/8" of the main hole, to (again) keep the larger diameter shank from splitting out a piece of the wood grain.
Don't over-tighten the swivel eye, as that will crush the surrounding wood.
I use a large allen wrench, threaded through the eye, as a swivel eye installation wrench, with bar soap on the screw threads (as posted above).
Practice your hole skills on some scrap wood, first.