The salutation "peace be with you" is very old indeed. It was commonly used by the Jews of Jesus' time and place with the Hebrew phrase "shalom" or "shalom aleichem" which is typically responded to with "aleichem shalom" which means "peace unto you." Muslims also have a similar phrase that is used as a common salutation with "as-salaam-alaikum" which means "peace be unto you." From primary sources around the early church we see this manifesting itself in Christianity as the kiss of peace.
As for its use and custom within the church we know that the holy kiss or kiss of peace was present in the Christian liturgy from the very early church. The following are some examples from New Testament Scripture:
- Romans 16:16—"Greet one another with a holy kiss"
- 1 Corinthians 16:20—"Greet one another with a holy kiss"
- 2 Corinthians 13:12—"Greet one another with a holy kiss"
- 1 Thessalonians 5:26—"Greet all the brothers with a holy kiss"
- 1 Peter 5:14—"Greet one another with a kiss of love"
Tertullian also wrote that before leaving a house, it was customary for Christians to exchange the kiss of peace and say "peace be to this house." In Augustine's Sermon 227 which was delivered over Easter, he explains the custom of the kiss of peace among the faithful. Like its place in the catholic liturgy today, we learn that by his time in the early church the kiss of peace was already part of the Eucharist after the Lord's Prayer.
So yes a salutation involving some form of passing or sharing peace was a common greeting in the early Christian church. However, it was not unique to Christianity and most likely came from the tradition of Jews to greet one another with "shalom" which was also incorporated into Islam which makes sense considering all of these faiths had periods of growth in the same part of the world.
As a side note, the mentions of peace coming from celestial beings is an on-going theme within the Old Testament and spills into the New. I think that this response is less about the greeting and more about a command to actually calm down and relax. The fact is that in the presence of God or angels we are struck with awe that probably leaves us paralyzed and terrified. Even the most faithful in the Bible require this brief call for calm from celestial messengers. I think this is something that is very powerful and present in the Bible and relevant for all Christians when thinking about prayer and the nature of God and His beings.