Begrudge shares its root with "grudge" and "grouch" — all born from medieval grumbling sounds, intensified by the English prefix be-.
To envy someone's enjoyment or possession of something. To give or allow reluctantly or resentfully.
From Middle English bigrucchen, from be- (intensive prefix) + grucchen (to grumble, complain), from Old French grouchier/grucier (to murmur, grumble), of uncertain ultimate origin, possibly Germanic imitative Key roots: grucchen (Middle English: "to grumble, complain"), grouchier (Old French: "to grumble, murmur").
Begrudge is built from the same root as "grudge" and "grouch" — all trace back to medieval grumbling. The be- prefix in English often intensifies the base word: bewilder (thoroughly wild), besmirch (thoroughly dirty), begrudge (thoroughly grudging). Old French grouchier may be onomatopoeic, imitating the low rumbling