Abarim Publications' online Biblical Greek Dictionary
The noun ηλιος (helios) means sun, and occurs a mere three times in the New Testament (Matthew 13:43, Mark 1:32 and Acts 13:11).
The Greeks had a god named Helios (Ηλιος) and the Romans had their Sol Invictus cult, but the New Testament appears to incorporate curiously little commentary on sun-worship. Still, see our article on the name Nazarene for a discussion on the competition between nationalistic solar cults and the message of personal freedom and responsibility of Jesus Christ.
Our noun appears to stem from a hugely ancient proto-Indo-European root from whence also stem the words sol (and hence solar) and sun. Equally old are the Latin words sollus (whole, entire), solus (alone, sole) and salus (being safe and sound; hence our word salvation), which share an etymological root in Sanskrit (says Lewis and Short's Latin Dictionary). In Greek these words are reflected by the related word ολος (olos), meaning whole, entire, complete, and although it's not specifically mentioned, here at Abarim Publications we wouldn't be surprised if the Latin words sol, solus, sollus and salus arose simultaneously in a most primitive solar worship and associated theology. It's even conceivable that the formations of the very old Semitic divine names El (in Greek Ηλ) and Elohim were either helped along or else directed by this root.
The Hebrew word for sun is שמש (shemesh), from whence comes the name Samson. Also note that the genitive of the name Elijah in Greek (Ηλιον), namely Ηλιου is the same as the genitive of the name Ηλιος.