The name Benique sounds French to me, like Véronique and Monique. They are Veronica and Monica in English, so I guessed that an English version on Benique might be Benica. I searched that and found a film director called Rob Benica. I also found a similar name, the Spanish Benicia, which means "blessed or benevolent one". The prefix ben- comes from Latin and means good, so I think Benique is maybe an elaboration of ben- in the French style, or it's a French version of Benicia. Hope that's helpful.
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