Firstly, it is significant that no women are listed among the greeters. This is in contrast to Rom 15:1-15, which mentions 27 people, of which 10 are women. The absence of women in Rom 16:21-23 is explicable if only travelers are listed, since women did not travel (see here).
Secondly, many of those listed are known to have travelled. Timothy had travelled extensively with Paul. I have argued that Jason was from Thessalonika and was Aristarchus, who was Paul's travel companion (Acts 19:29; 20:4). Sosipater was Sopater, another traveller (Acts 20:4). Gaius was, I have argued, Stephanas, who had been to Ephesus on church business. Erastus was the Erastus who had travelled around the Aegean with Timothy (Acts 19:22).
This analysis shows that Lucius too must have travelled among the churches. Indeed, he is mentioned second only to Timothy, and his prominence in the list suggests that he had been a more prolific traveller than the others (Jason, Sosipater, Gaius, Erastus and Quartus). But how widely had he travelled? I suggest that a clue can be found in 2 Cor 8:18-19 where we read of a "brother" who
"is famous among all the churches for his proclaiming the good news; and not only that, but he has also been appointed by the churches to travel with us while we are administering this generous undertaking for the glory of the Lord himself and to show our goodwill"This brother had been appointed to accompany Paul to Judea so he was almost certainly with Paul when Paul wrote Romans just before leaving for Judea. He was well known in "all the churches" so he was presumably known to many who had moved to Rome. It is very likely, therefore, that he is one of those who sent greetings in Rom 16:21-23. He cannot have been Timothy, not least because Timothy was Titus. Therefore he was either Lucius or he was one of those mentioned after Lucius (Jason, Sosipater, etc.). Therefore Lucius was either the man who was famous among the churches, or he was even more prominent than him.
He must surely have been the author of Acts because:
1. It is unlikely that Acts would have failed to mention one who travelled among the churches so extensively.
2. The author of Acts would almost certainly have sent greetings to the church of Rome since he would have known many of them through his extensive travel, and he was with Paul in Corinth at the time that Romans was written since he planned to travel with Paul to Judea (Acts 20:2-5). Ro 16:21-23 contains a complete list of the prominent believers who were in Corinth at the time. See also my discussion here.
3. The name "Lucius" is the full form of the name "Luke", which the church fathers unanimously attach to Luke-Acts.
In Philemon 23-24 Paul sends greetings from Epaphras, Mark, Aristarchus, Demas and Luke. Again there are no women among the greeters, which suggests that Paul is here sending greetings from those who knew Philemon from visiting his town. This is supported by the fact that Aristarchus frequently travelled on church business (Acts 19:29; 20:4; 27:2). Epaphras was also a church envoy if I am right to equate him with Ephaphroditus (see here). For what it is worth, the disputed letters also assume that Epaphras (Col 4:12), Mark (Col 4:10; 2 Tim 4:11), and Demas (2 Tim 4:10) were traveling co-workers of Paul.
I think we can be confident, therefore, that Luke was a traveling co-worker of Paul. This point tightens the argument for equating him with the Lucius of Rom 16:21 and the author of Acts.
For more on why the author of Acts = Luke = Lucius, see my earlier posts here, here and here.