schon - yet, already
schön - well, nice, pretty, beautiful
Wiktionary has schon from an old German word equivalent to modern schön. I think this is typologically unlikely, though of course my sample base is too small to be sure. I only see pretty early, reflecting the fact that pretty didn't always mean what it primarily denotes today.
Specifically, I rather want to compare it to soon, which matches idiomatically "ich werde das schon-/-noch machen" (which is difficult to translate, circa in due time), though the semantic burden could appear to rest on noch "still", depending on which part of the collocation is elided; it's chiefly schon in "wir werden das kind schon schaukeln" (we will well drive the point home)
: noch is also difficult to translate and etymologically without compare; probably from *h2nek- "to reach, attain", cp. enough, genug, also next, nachher "after this'ere"), where DWDS.de sees an influence of *ne, too.
Perhaps an influence of schön could be seen in schon gemacht "already did", which carries a connotation of affirmation, cp. schön gemacht "well done". schön instead appears akin to show (think, nice view).
Whether schon could be a palatilizing dialectal variant of MHG san, son (no gloss, but cog. with soon, possibly from deictic *so/to-, or perhaps *swe-) and vice versa schön related to Fr. jeune is not in question, only the typology schön > schon.
Note of caution: If it's not likely, that might simply mean that typology is not good meassure in this case, which might mean that the development were overall too complicated to be frequent, to be found out easily.