I believe it has to do with the verb cligner [des yeux] (to blink), which then gives the noun clin.
As @MorganFR pointed out, it is possible that it could refer to something else in a specific field (such as boat building).

Larousse also says “Biseau formé à l’extrémité de chacune des douelles du tonneau”.

Which is “Bevel formed at the end of each of the cask staves”.

I just learned it as I answer.

There are two homonyms spelled clin.

One is common and only used in the clin d’œil / clin d’yeux expressions. It comes from cligner.

The other one is quite rare and belongs in the ship building and similar vocabulary and means some kind of overlapping tiles. It comes from the Dutch klinken, meaning riveter/boulonner (to rivet/to bolt).