Romans 16:21-24 meaning

Some of those with Paul in Corinth sent personal greetings to the believers in Rome. We know much about Timothy from the rest of Paul's letters. Paul was Timothy's mentor and eventually Timothy served as Paul's representative to several churches, teaching and instructing them. 

Presumably, Timothy was with Paul when he finished this letter and sent his greetings. Timothy my fellow worker greets you, and so do Lucius and Jason and Sosipater, my kinsmen (v 21), meaning they are of Jewish origin. A Lucius is also mentioned in Acts 13:1 as a teacher of the church at Antioch; this could be the same person. A Jason is also mentioned in Acts 17:1-10, he hosted Paul and Silas at Thessalonica. Because he hosted them, an angry mob attacked him, but he was eventually released by the authorities; this could be the same Jason. The Sosipater mentioned here is probably the same man who accompanied Paul on his third missionary journey (Acts 20:4).

We do not know anything else about Tertius besides what we learn here: I, Tertius, who write this letter, greet you in the Lord (v 22). He was the scribe for this letter. Paul might have dictated the letter to him because of his failing eyesight (Galatians 6:11). 

Gaius, host to Paul and to the whole church, greets you (v 23). Gaius hosted Paul while he stayed in Corinth and evidently also hosted a church at his home. This is the same Gaius whom Paul baptized in 1 Corinthians 1:14. It is unclear whether or not this is the same Gaius mentioned in Acts 19:29, Acts 20:4, or the Gaius that the third letter of John addresses. Some scholars say some of these reference the same person, some say these are four different people. 

Erastus also sent greetings and was Corinth's city treasurer (v 23), an esteemed position. An Erastus is also named as a man who accompanied Timothy to Macedonia (Acts 19:22) and is mentioned again in salutations to Timothy (2 Timothy 4:20). 

It is once more unclear if these are all references to different people or the same. All we know about Quartus is written here, that he was a brother in Christ (v 23). The blessing given in this section is in parentheses because it is only found in some manuscripts and is not considered to be legitimate by some scholars: The grace of our Lord Jesus Christ be with you all. Amen (v 24).

This is the twenty-fifth occurrence of the word translated grace in this letter. Again, this underscores Paul's emphasis that the gospel of Jesus Christ is a gospel of grace. 

It is evident from the greetings Paul and his companions sent that they greatly cared for the believers in Rome. In Romans 12, Paul described the church as a body with each member of the body doing their part and caring for other believers. These believers cared about each other from afar and truly sought harmonious living amongst each other.