Abarim Publications' online Biblical Greek Dictionary
The noun υιος (huios) is the common Greek word for son (Matthew 1:21, Mark 1:19, Luke 1:13, John 1:42), but where the English word "son" simply denotes one's male child, in Greek this word covers a far wider range of relationship and a much narrower range of progeneration.
Our word υιος (huios) is the masculine counterpart of the feminine noun θυγατηρ (thugater), meaning daughter, but the opposite of νοθος (nothos), which denotes a bastard or illegitimate child. That means that our word doesn't simply denote one's male biological offspring, but rather a person who has been acknowledged and accepted into a household as someone who has the rank and rights of a male offspring, and whose destiny it is to represent and ultimately replace the male parent upon his death.
But it's the final clause that most defines our word. Unlike our English word "son", which brings to mind the mere biological act of procreation and is an epithet that can never be removed, the Greek word υιος (huios) brings to mind the political act of representation, which could be bestowed and rescinded independently from any physical affiliation.
What may seem like a figurative use in English, in Greek it's well within our word's literal applicability: the use of the word "son" to describe a member of some specific group, such as physical descendants over multiple generations (of Abraham: Matthew 3:9; of Israel: Matthew 27:9; of Levi: Hebrew 7:5; of David: Matthew 9:27) but also of a craft, skill, duty or aptitude (of a kingdom: Matthew 13:38; of the bridal chamber: Matthew 9:15; of thunder: Mark 3:17; of this age: Luke 16:8; of light: John 12:36; of the prophets and of the covenant: Acts 3:25; of disobedience: Ephesians 2:2; of destruction: 2 Thessalonians 2:3).
Hence, our word is also used to describe an acolyte or apprentice (of the Pharisees: Matthew 12:27), as its equivalent τεκνον (teknon), meaning child (2 Timothy 1:2), and which literally means "production".
It's in this same sense that a human person, or man-child, is often referred to as "son of man" (Mark 3:17, Ephesians 3:5), although Jesus' title of "Son of Man" obviously denotes a new human race that derives from the natural human race; son of humanity. But it's in the sense of reproducing the character of the Father that Jesus is called Son of God (Matthew 14:33), and the same goes for all the other "sons of God" mentioned in the Bible:
- Adam (Luke 3:38)
- The Pre-flood Pack (Genesis 6:2)
- Israel (Exodus 4:22, Hosea 11:1); the Israelites (Isaiah 1:2, Hosea 1:10); Ephraim (Jeremiah 31:9)
- The Angelic Earth Patrol (Job 1:6)
- The pre-incarnate Messiah (Psalm 2:12)
- Everyone called by the Lord's Name (Isaiah 43:6-7)
- Peace makers (Matthew 5:9)
- The resurrected (Luke 20:36)
- Those led by the Spirit of God (Romans 8:14-19)
- Those who have faith in Christ Jesus (Galatians 3:26)
Where our noun υιος (huios) comes from isn't wholly clear, but to a poetic Greek there may have seemed to be some associative similarities with the verb υω (huo), to rain, from whence comes the familiar words υδορ (hudor, hence "hydra"), or water — our word's opposite, namely νοθος, nothos, looks like νοτιος (notios), meaning moist or pre-rain.
These words in turn may seem somewhat reminiscent of the verb υδεω (hudeo), to call or name, of which the various spellings in turn have some visual overlap with the verb οιδα (oida), to know. A Greek with a Hebrew background would, of course, remember that the Hebrew word חפר (haper) means both dry and shame, whereas the word מורה (moreh), means either rain or teacher (and is closely related to the word תורה, or Torah).
A Hebrew would also remember that in his native language the word for son, namely בן (ben) looks like it comes from the verb בנה (bana), to build, and even has to do with the word אבן ('eben), or stone. The feminine version of בן (ben), meaning son, is בת (bat), meaning daughter, which has a striking resemblance to the word בית (bayit), meaning house.
Our noun comes with one derivative, namely together with the magnificent verb τιθημι (tithemi) meaning to set, put or establish: the noun υιοθεσια (huiothesia), meaning adoption (Romans 8:15-9:4, Galatians 4:5, Ephesians 1:5). Note that in our modern world we may distinguish a "real" son from an "adopted" son on account of physical descent, but in Greek that doesn't work that way. An adopted son was in every way a "real" son, and someone who actually physically descended from a man would not automatically be his "real" son.