No. Two different people called Herodion.
Herodion in Romans
Now Paul mentions a man named Herodion. Since this would have been an unusual name in Rome, scholars suggest this may have been either a family member or former slave who served one of the royal lines of the Herods. Since Paul calls him a kinsman or relative, it is assumed the man is Jewish.
— "What does Romans 16:11 mean?", BibleRef
Which we find on Wikipedia as (St) Herodion of Patras:
Ἡρωδίων, Ἡρωδιανός, Ῥοδίων […]
According to tradition, he was numbered among the Seventy Disciples and became bishop of Patras, where he suffered greatly. After beating, stoning, and stabbing him, they left him for dead, but St. Herodion arose and continued to serve the Apostles.
He was beheaded with Olympas in Rome while they were serving Saint Peter on the same day that St. Peter was crucified.
Saul in Josephus
The key to sort this Herodion into a proper family tree with Herodes in it is Costobarus:
This is then further explained clearly:
Costobar and Saul were royal Herodian brothers, and kinsmen of Antipas ben Alexas, and of Agrippa II. While Josephus does not specify the parents of Costobar and Saul, the name “Costobar” provides a clue: their grandfather was very likely Costobar(us), the second husband of Salome, the sister of Herod “the Great”. “Costobar” is an uncommon name, there being only these two individuals so named in all of Josephus' writings. This means that Antipas ben Alexas was a nephew of Costobar and Saul, through their sister Cypros bat Antipater and Agrippa II was first cousin once removed to them through their aunt Bernice (Berenice), who married Aristobulus ben Herod and are the parents of Agrippa I.
When the Jewish insurrection against Roman rule was gaining momentum, Costobar, Saul and Antipas requested Agrippa II to send assistance to prevent the imminent uprising. The two brothers also were active against the insurgents.
Following the Battle of Beth-horon (25 November 66 AD) in which the Jewish insurgents defeated the Roman general Cestius, Costobar, Saul and Antipas were besieged in the royal palace. Subsequently, Costobar, with his brother Saul, escaped from Jerusalem to re-join Cestius: who dispatched them to Emperor Nero in Archaia, Greece.
Antipas, who had remained in Jerusalem, was arrested by the insurgents, and slain in prison by John ben Dorcus, (i.e. John benTabitha), who was under commission from the “brigands” [zealots].
There is a fringe theory that the Saul in Josephus' writings was the same person as Saul of the New Testament. According to this theory, references to Saul in Acts of the Apostles and some verses of Paul's Epistle to the Romans are believed to reveal connections to the Herodian royal family. In Acts of the Apostles, Saul is named in a list of Christian prophets and teachers in Antioch, following Manaen, who was “brought up with Herod the Tetrarch”, but the verse does not clearly connect Saul to Manaen, or to Herod; In the last chapter of the Epistle to the Romans, Paul sends greetings to a man named Herodion, whom he calls a kinsman (Gk: συγγενῆ). However, Herodion is a not uncommon name in the ancient world, and Paul refers to several others as kinsmen in the same chapter. This term likely meant nothing more than that they were also Jewish. If he were a member of the Herodian family, Saul would indeed have been a Roman citizen. His behaviour prior to his conversion, in which he “made havoc for the church” could be seen as reminiscent of that in which Costobar and Saul “were lawless and quick to plunder … those weaker than themselves”, however, the account of the violent behaviour of Costobar and Saul in Josephus would have post-dated Paul's conversion to Christianity by decades.