עזז

Abarim Publications' online Biblical Hebrew Dictionary

עזז ענז

The forms עזז ('zz) and ענז ('nz) have nothing to do with each other etymologically, but both produce words that are spelled עז ('z). Then there is the root עוז ('uz), which possibly is related to עזז ('zz):

The Hebrew verb עזז ('azaz), generally means to be strong or powerful. This verb is often applied to denote one of God's inherent qualities (Psalm 89:13) compared to man's weakness, and when it's applied to man, it usually denotes either a prevalence in battle (Judges 3:10) or arrogance (Proverbs 21:29). The Preacher notes that wisdom is strong for the wise (Ecclesiastes 7:19).

This verb yields the following derivatives:

The adjective עז ('az), meaning strong, mighty or fierce (Numbers 13:28, Judges 14:18). This word is never applied to God.
The masculine noun עז ('oz), meaning strength (Isaiah 26:1, Psalm 8:2). This word is used primarily for God, particularly in the Psalms.
The masculine noun עזוז ('ezuz), meaning strength, might, fierceness (Isaiah 42:5, Psalm 145:5).
The adjective עזוז ('izzuz), meaning mighty or powerful (Isaiah 43:17, Psalm 24:7).
The feminine noun עזניה ('ozniya), denoting some kind of bird of prey; a vulture or osprey (Deuteronomy 14:12, Leviticus 11:13). This curious word is probably imported from another language.

ענז

The root ענז ('nz) isn't used in the Bible but it occurs all over the Semitic language spectrum with meanings relating to wayward motions or a turning aside (perhaps even to be strong-headed?). Its sole derived Biblical noun occurs in other languages as well, all with meanings similar to the Biblical one: the feminine noun עז ('ez), meaning she-goat.

She-goats could be utilized as unit of wealth (Genesis 30:32), were used for food (Genesis 27:9), or for sacrifices (Leviticus 22:27). The hair of the she-goat could be used to fabricate items such as blankets (1 Samuel 19:16), and the sight of his lover's flowing hair reminded king Solomon of a flock of she-goats descending from Mount Gilead (Song of Solomon 4:1). Note that cattle were units of wealth and wealth yields strength in a social sense.

עוז

The verb עוז ('uz), meaning to rush or bring into refuge or to seek safety, is possibly related to the verb עזז ('azaz), meaning strong. It occurs a mere half a dozen time (Exodus 9:19, Isaiah 30:2, Jeremiah 4:6).

This verb's sole derivative is the masculine noun מעוז (ma'oz), meaning place or agent of safety or protection. It may describe a literal place of safety; a stronghold (2 Samuel 22:33, Ezekiel 24:25, Daniel 11:7), or the protectorate of a human person (Isaiah 30:2), or a helmet (Psalm 60:7), or an angel (Daniel 11:1). But mostly it describes the protectorate of YHWH (Nehemiah 8:10, Psalm 27:1, Isaiah 25:4, Nahum 1:7).