How did the New Testament writers view Jesus' title of "Son of God". What did the early church believes about what it means?
"Messiah" ("Christ" in Greek) is a Hebrew word that essentially means the anointed one. Prior to the crucifixion, the expectation of both Jews and Christians was that a man would arise who would be reminiscent of David, whom Samuel anointed to be king. The Messiah would deliver Israel from oppression and lead her as their king.
The NT authors and the early church saw Jesus as the Messiah; He was the deliverer from spiritual oppression, and the king of a spiritual kingdom. He would some day deliver them from physical oppression, and rule a physical kingdom.
"Son of God" essentially meant God's image bearer. The idea of a "son" in the first century was different than it is today. The term "Son of God" did not so much signify that Jesus was born, or that He had a beginning, or that He was somehow inferior to the Father, but rather that He came from God and bore His image. (cf)
Are the two synonymous?
Yes and no. They signify different things by definition, but in the Person of Christ they were seen as equally valid and important titles, and are often treated as if they were synonymous. Here is a brief sampling of places where we see this: