The name Bebai in the Bible
There are two men named Bebai in the Bible, or so it is reasonably assumed. Ezra mentions a patriarch whose 623 "sons" were among those who joined Zerubbabel in the second wave of Israelites to return to Canaan from the Babylonian exile. It's generally regarded safe to assume that patriarch Bebai wasn't alive to see 623 of his male offspring, and so the Bebai who Nehemiah mentions among the signers of the sealed document, is another Bebai (10:15).
Etymology and meaning of the name Bebai
It's unclear where the name Bebai comes from, but one of the candidates is the curious word בבה (baba), meaning "apple" of the eye or pupil. This feminine noun occurs in the Bible only in Zechariah 2:8 (in the construction "the baba of the eye", and the etymology of this word is uncertain. Some scholars (Gesenius, for instance) propose relations with the Hebrew verb נבב (nabab), meaning to hollow out (Job 11:12, Jeremiah 52:21), and translate our word בבה (baba) with "opening" of the eye. Others suggest it has to do with a child's babble (that may also have given us the word baby) and translate our word baba with "child" of the eye.
NOBSE Study Bible Name List translates the name Bebai with Fatherly, which apparently means that a relation with the word אב (ab), meaning father, is proposed:
Alfred Jones (Dictionary of Old Testament Proper Names) believes our name Bebai is a contraction of (a) the particle ב (be), meaning in our by:
(b) The root of the word אב ('ab), meaning father, which is אבה ('aba), which for this occasion is said to be a verb and to mean to desire or long for (an interpretation which is not shared with any of the other sources, and is dubious at best); and (3) a truncation of יהוה (YHWH; Yahweh, the name of the Lord). Hence Jones renders the name Bebai the meaning of With The Desire Of The Lord.