What does Gideon mean?
From the Hebrew verb gada, 'to hew down, cut off,' hence 'hewer' or 'feller' — fitting the judge who cut down Baal's altar (Judges 6). The common 'mighty warrior' gloss is a secondary association from his role, not the meaning of the root; because no noun is formed from the verb, the exact nuance stays somewhat uncertain.
What does Ari mean?
From Hebrew ari (also aryeh), 'lion.' A short modern Hebrew name; distinct from the theophoric Ariel ('lion of God'). A concrete noun — its strength associations are connotation, not etymology.