His father was Pedaiah.
17 The sons of Jeconiah, the prisoner, were Shealtiel his son,
18 and Malchiram, Pedaiah, Shenazzar, Jekamiah, Hoshama and Nedabiah.
19 The sons of Pedaiah were Zerubbabel and Shimei.
1 Chronicles 3:17-19 NASB
His father was Shealtiel. (5 references)
1 Now when the seventh month came, and the sons of Israel were in the cities, the people gathered together as one man to Jerusalem.
2 Then Jeshua the son of Jozadak and his brothers the priests, and Zerubbabel the son of Shealtiel and his brothers arose and built the altar of the God of Israel to offer burnt offerings on it, as it is written in the law of Moses, the man of God.
Ezra 3:1-2 NASB
Problem
The question is: How is it possible that Zerubbabel is named (1) a son of Pedaiah, and (2) a son of Shealtiel?
Preparation for a solution
In the first place genealogies often begin with the standard opening formula: These are the sons of, or: the sons of. A more convenient translation would be: These are the descendants of, or: the descendants of. E.g. the b’ne Israel are the sons of Israel (Jacob), the children of Israel, or the descendants of Israel, Israelites (Exodus 1:9 etc.).
Take this into account in 1 Chronicles 3:17-19, and we get:
17 The descendants of Jeconiah, the prisoner, were Shealtiel his son,
18 and Malchiram, Pedaiah, Shenazzar, Jekamiah, Hoshama and Nedabiah.
19 The sons of Pedaiah were Zerubbabel and Shimei.
Comments
The answer can be short and clear. In verse 17 we read now that Jeconiah had only one son, which is understandable as he was for long time (37 years, 2 Kings 25:27) a prisoner in Babylon without family life.
The names in verse 18 are now all sons of Shealtiel and grandsons of Jeconiah.
In verse 19 we read about Zerubbabel as a son of Pedaiah. That means Zerubbabel is a grandson of Shealtiel. According to the rule a son of a son is a son, it is absolutely correct to speak of Zerubbabel the (grand)son of Shealtiel.
No Bible Contradiction
See also the article 21-30 of 101 Islamic Bible Contradictions? No. 28, 29.