Although the noun euaggelion (“good news, gospel”) and its cognate verb euaggelizomai (“communicate good news”) occur frequently in the New Testament, the related term euaggelistēs (“one who proclaims good news”) only appears three times (Acts 21:8; Eph. 4:11; 2 Tim. 4:5). No one applied this term to Jesus. It does not appear in Greek literature before the New Testament writings. It may occur in one inscription from the island of Rhodes (IG, XII, 1, 675, 6), but the nature of the inscription and the meaning of the term, as well as its date is disputed. Liddell, Scott and Jones (Greek-English Lexicon (1966), 705) suggest its meaning in the Rhodes inscription is “proclaimer of oracular messages.” Horsely (New Documents Illustrating Early Christianity 3, (1983), 14) agrees with the meaning offered by Liddell, Scott and Jones, and is inclined to think that the inscription is non-Christian. Another cognate adjective εὐάγγελος “bringing good news,” is an epithet applied to the god Hermes, the divine messenger.
According to R. E. Van Voorst (Building Your New Testament Greek Vocabulary (1990), 9), nouns formed with the suffix -tēs typically signify a person or agent (cf. Robert Funk, A Greek Grammar of the New Testament, 1961, 59), e.g., baptistēs (“baptizer”), proskunētēs (“worshiper”). Based on these analogies euaggelistēs probably means an “agent who proclaims a message about significant news.” The source and nature of the message would be determined by the context.
Paul’s usage of this noun in Ephesians 4:11 is the first occurrence chronologically in the New Testament (c. 60 AD, during his Roman imprisonment). He lists euaggelistai among the Spirit-endowed people Jesus gifts to his church as resources. These include “apostles, prophets…shepherd-teachers.” They “equip holy ones (believers) for any task that assists” the Messiah fulfill his mission. Paul defines this outcome in Eph. 4:16 — so that the whole body can grow the body. In this context a euaggelistēs works among believers to train and disciple them. This suggests that they work with believers rather than non-believers. According to Paul’s description in Eph. 4:13-15 these gifted people keep the community of believers grounded in the gospel, able to deal with false-teaching and other matters that would discourage or deceive them. They “speak the truth in love” (Eph. 4:15), helping believers become mature disciples (cf. 4:20-24). If we apply the sense of its usage in the Rhodes description, as proposed by Liddell, Scott and Jones, then such people would be communicating the teaching of Jesus, i.e., the good news that he proclaimed (Mark 1:14-15) about the Kingdom of God. Perhaps these people had the capacity to retell the Jesus traditions in the churches, such as we find in the written Gospels, before there were written Gospels. And in addition to instruct believers in the ways that the Jewish Scriptures, the Christian Old Testament, came to fulfillment in Jesus Christ.
Presumably Paul writes his second letter to Timothy around 63-64 AD. in 2 Tim. 4:5 he instructs his protege, Timothy “Do the work of an euaggelistou.” In some way this is part of his ministry assignment that he must fulfill. This command comes at the end of a long list of instructions (4:2-5) that begins with Paul urging him to “proclaim the message” (v. 2). This “healthy teaching” is necessary if the Ephesian church is to hold fast to the true message in the face of false teaching. Again, the focus is upon equipping believers to stand firm when confronted with deceptive ideas. Perhaps “doing the work of an evangelist” is the same as “proclaiming the message,” even if it generates opposition and suffering.
The third usage comes in Acts 21:8. Acts is the second part of the massive Luke-Acts composition in the New Testament. It probably postdates Paul’s letters to the Ephesians and to TImothy. However, Luke was one of the early church leaders associated with Paul, particularly in the later part of his ministry when he was imprisoned, i.e., when he wrote the letters to the Ephesians and to Timothy (cf. the mention of Luke in 2 Tim. 4:11). Paul, on his way to Jerusalem, accepts hospitality in Caesarea from a person named Philip, one of the seven selected by the Jerusalem church to oversee the care of widows (Acts 6:5). In Acts 8:5 the writer recounts how Philip went to Samaria to “proclaim (εκēρυσσεν) the Messia.” However, the verb ευαγγελιζομαι describes this work of Philip in Samaria and also in discussion with the Ethiopian Eunuch (8:4, 12, 35, 40). This same term describes the actions of Peter and John (5:42; 8:25), Jesus (10:36), believers in Antioch (11:20), and Paul and Barnabas (13:32; 14:7). So this kind of activity is widespread among the followers of Jesus in the early church. However, among them in Acts only Philip is called an euaggelistēs 14:8. Unfortunately, the narrative in Acts does really help us discern exactly what function this term describes.
I think we have to rely on Eph. 4:11 and 2 Tim. 4:5 to discern what this term means in the context of the early church. A person identified with this term would have the capacity to present the message about Jesus and his teachings in ways that refuted false teaching. Positively, such people would promote the validity of the gospel message by showing how the Old Testament scriptures were fulfilled by Jesus. Perhaps also they could transmit the oral record of Jesus’ miracles and teaching.