The only other modern word I know of to employ the same root actually stems from “valediction” itself.
"student who pronounces the oration at commencement exercises of his or her class,"
1832, American English
Source: etymonline.com definition of “valedictorian”
“Valē” is the 2nd-person singular imperative of “valēre”, which means “to be well”. English has done away with most declensions such as this, so now it's “farewell” whether you’re saying it to one person or a ham.
UPDATE: I found a list of words derived from “valēre”:
ambivalence, ambivalent, avail, countervail, convalesce, equivalent, equivalence, evaluate, evaluation, evaluator, invalid, invalidate, overvalue, overvaluation, prevail, prevalent, prevalence, reevaluate, reevaluation, undervalue, undervaluation, valance, valence, valency, valediction, valetudinary, valiance, valiant, valid, validate, validation, validity, valor, valuable, valuation, value
They may not all seem very related, but remember that one sense of “well” is basically “strong” and indeed “valē” is purported to descend from the Proto-Indo-European *wal (“be strong”).¹