Godspeed is recorded as a two-word phrase (God speed) in the King James New Testament. The phrase is only found in 2John, a single chapter letter John wrote to a Christian woman and her family. The epistle, written between 63 and 64 A.D., both encourages the Christian family but also warns them not to wish "Godspeed" to those preaching false doctrine.
The Greek behind Godspeed, found in 2John 1:10 - 11, is chairo (Strong's Concordance #G5463). It is only translated as this word in the King James Version Bible. Other Bibles avoid using Godspeed, opting instead to use "Welcome" (HCSB, NIV, HBFV), "greet" (NKJV), "greeting" (NASB, ASV) and "Hail!" (YLT).
If there come any unto you, and bring not this doctrine, receive him not into your house, neither bid him God speed (chairo): For he that biddeth him God speed (chairo) is partaker of his evil deeds (2John 10 - 11, KJV).
Why the warning?
John, before addressing the issue of using Godspeed, first warns his readers that deceivers (antichrists) are promoting the belief that Jesus did not come to earth as a flesh and blood human (2John 1:7). This deception, a gnostic heresy, is known as Docetism (see 1John 4:2 - 3).
Gnostics believed that matter was inherently evil but that spirit was good. Based on this they promoted the teaching that Jesus only seemed to be human but was actually a spirit being (A Zeal for God Not according to Knowledge by Eric Snow, Section 6). Their nutty belief that Christ was not God manifested in the flesh is in direct contradiction to many Scriptures (Luke 1:30 - 35, Philippians 2:6 - 11, 1Timothy 3:16, etc.).
One major way this heresy spread was through itinerant "preachers" who presented themselves as mature Christians who served God. When they entered an area, they would commonly rely on local people for support and hospitality. Their teachings, however, brought negative spiritual consequences to those who adopted them (2John 1:8). John warned to not board these individuals or even wish them Godspeed (wish them well, ask God to bless their efforts, etc.).
This, of course, does not mean true Christians should wish someone ill will or curse them out as a replacement for wishing them Godspeed. Believers should always be courteous and harmless like a dove (Matthew 10:16). They are not required, however, to offer non-emergency aid or offer a blessing to others when they are seeking to subvert the truth of God.