The name Eliza is a Hebrew baby name. In Hebrew the meaning of the name Eliza is: or Elizabeth, from Elisheba, meaning either oath of God, or God is satisfaction. Also a diminutive of Bethia (daughter or worshipper of God), and of Bethany, a New Testament village near Jerusalem.he name Eliza is an English baby name. In English the meaning of the name Eliza is: My God is bountiful. God of plenty. A frequently used as an independent name. Famous bearer: Eliza Doolittle, the Cockney heroine of George Bernard Shaw's play 'Pygmalion', from which the musical 'My Fair Lady' was derived.