This use of Palm branches was common among ancient Greeks, Romans and Hebrews. It seems from the Hebrew use, palm branches symbolized ’rejoicing’ during the Feast of Tabernacles. During the feast Hebrew people were commanded to take branches of palms, with other trees, and rejoice before God.(Leviticus 23:40) However this was just not palm branches, but other ‘leafy branches’ as well, along with fruits. Later on the specific use of palm branches also agrees closer to with the meaning used in Greece, for example in 142 BC, when Simon Maccabees liberated Jerusalem from the Greeks:
On the twenty-third day of the second month, in the year 171,[c] there was a great celebration in the city because this terrible threat to the security of Israel had come to an end. Simon and his men entered the fort singing hymns of praise and thanksgiving, while carrying palm branches and playing harps, cymbals, and lyres. (1 Maccabees 13:51)
Before the time of the Maccabees, Greek Hellanodikai ( judges at the Olympics) awarded winners with ‘palm branches’ and crowns (source).
Most vivid and powerful was the waving of palm branches by the Romans as they celebrated military successes with palm branches and most relevant to Palm Sunday, welcomed royalty by waving them.
Some times the best historical records are printed on coins as they do not deteriorate as much as paper. According to ancient coins:
Palm fronds had been used for centuries before the Romans as an attribute of winged Νικη (Nike), the goddess whose name is the Greek for Victory. The Hellenic bronze coin on the near right is from Sinope, 120-63 BCE, and shows Nike carrying her palm frond over her shoulder. Her name has been anglicized into a well-known brand of sportswear. (source)
Waving and use of palm branches during the time of Christ was a multicultural symbol symbolizing victory, praise and the presence of royalty. It also shows us that although God speaks to any given people within their culture and that His word condescends into it, speaking in familiar ways, the culture itself does not have to have any ‘deeper meaning’ beyond what it meant at the time. With a basic understanding of historical culture, Christ’s triumphant entry into Jerusalem when greeted by throngs of people making a make-shift carpet for him out of the branches and coats, or in Revelation where the saints are holding them before the throne. It simply meant he was being treated as the 'future King' in the gospels and as the 'present king' in revelation.