My Oxford Dictionary of English Place-Names has this entry: "Solent, the Hants. Soluente 731. An ancient pre-English name of uncertain origin and meaning."
The adjective, according to the online OED, means "usual" or "customary," and the etymology is stated as: "[? f. L. sol-{emac}re to be wont.]" This is their quotation entry: "1658 FRANCK North. Mem. (1694) p. vi, There the Rocks and the Groves will be our solent Reception, and the Cities and Citadels supply us with Accommodation."