In a Sixth Century book called Liber Pontificus, or Lives of the Popes, and repeated in Bede’s History of the Church in England, St. Lucius was the first Christian king of Britain, and, we note the first King anywhere to convert, making this the first example of the Christian King or the question of Christianity and kingship. The Emperor Constantine did not convert Rome until the Fourth Century. According to the story in Wikipedia:
The story became widespread after it was repeated in the 8th century by Bede, who added the detail that after Eleutherius granted Lucius’ request, the Britons followed their king in conversion and maintained the Christian faith until the Diocletianic Persecution of 303.
The English monk Bede included the Lucius story in his Ecclesiastical History of the English People, completed in 731. He may have heard it from a contemporary who had been to Rome, such as Nothhelm. Bede adds the detail that Lucius’ new faith was thereafter adopted by his people, who maintained it until the Diocletianic Persecution. Following Bede, versions of the Lucius story appeared in the 9th-century Historia Brittonum, and in 12th-century works such as Geoffrey of Monmouth‘s Historia Regum Britanniae, William of Malmesbury‘s Gesta Pontificum Anglorum, and the Book of Llandaff. The most influential of these accounts was Geoffrey’s, which emphasizes Lucius’ virtues and gives a detailed, if fanciful, account of the spread of Christianity during his reign. In his version, Lucius is the son of the benevolent King Coilus and rules in the manner of his father. Hearing of the miracles and good works performed by Christian disciples, he writes to Pope Eleutherius asking for assistance in his conversion. Eleutherius sends two missionaries, Fuganus and Duvianus, who baptise the king and establish a successful Christian order throughout Britain. They convert the commoners and flamens, turn pagan temples into churches, and establish dioceses and archdioceses where the flamens had previously held power. The pope is pleased with their accomplishments, and Fuganus and Duvianus recruit another wave of missionaries to aid the cause. Lucius responds by granting land and privileges to the Church. He dies without heir in AD 156, thereby weakening Roman influence in Britain.
Later traditions are mostly based on one of these accounts, probably including a medieval inscription at the church of St Peter-upon-Cornhill in Cornhill, London in the City of London. There, he is credited with having founded the church in AD 179.
Bede, in the fourth chapter of the first book of his History, writes:
In the year of our Lord’s incarnation 156, Marcus Antonius Verus, fourteenth from Augustus, became emperor jointly with his brother Aurelius Commodus. During their reign, and while the Holy Eleutherus ruled the Roman Church, Lucius, a British King, sent him a letter, asking to be made a Christian by his direction The pious request was quickly granted, and the Britons received the faith and held it peacefully in all its purity and fullness until the time of the Emperor Diocletian.
And from the Liber Pontificus, listing the oldest listing of the Bishops of Rome:
XIV . ELEUTHERIUS
Eleuther, by nationality a Greek, son of Habundius, from the town of Nicopolis, occupied the see 5 years, 3 months and 2 days. He was bishop in the time of Antoninus and Commodus until the year when Paternus and Bradu a were consuls (AD 185). He received a letter from Lucius, king of Britain, asking him to appoint a way by which Lucius might become a Christian. He also decreed He also confirmed again the decree that no kind of food in common use should be rejected especially by the Christian faithful, inasmuch as God created it; provided, however, it were rational food and fit for human kind He held 3 ordinations in the month of December, 2 priests, 8 deacons, 5 bishops in divers places. He also was buried near the body of the blessed Peter in the Batican, May 24. And the bishopric was empty 15 days.
As to the meaning of the Greek name of Pope Eleutherius, Google answers:
The Greek word “ἐλευθερία” (capitalized Ἐλευθερία; Attic Greek pronunciation: [eleu̯tʰeˈria]), transliterated as eleutheria, is an Ancient Greek term for, and personification of, liberty. … In Ancient Greece, Eleutheria was also an epithet for the goddess Artemis, and as such she was worshipped in Myra of Lycia.
Nennius, about 809,  writes:
Lucius, the British king, received baptism, with all the underkings of the British nation, 167 years after the coming of Christ, after a legation had been sent by the roman emperors and by Eucharistus, the Roman Pope.
According to Geoffrey of Monmouth, Lucius died without an heir, and the rule of Britain was quickly usurped before Severus Restored the rule of Rome in the early 200’s. Oddly, Geoffrey refers to Gildas , though Gildas seems to leave out the story of Lucius. If Britain was converted in 156, and Bishoprics established by 176, we would have about one century, three or four generations, to connect the Cole line of Lucius to the Colchester father of St. Helen.