What is the meaning of God's omni-powers?

Here's a few verses to add to what has been said.

Omnipresence:

 "He is before all things, and in him all things hold together"
 Colossians 1:17
 "For in him we live and move and have our being." 
 Acts 17:28
 "Very truly I tell you," Jesus answered, "before Abraham was born, I am!"
 John 8:58
 "But who is able to build a temple for him,
 since the heavens, even the highest heavens,
 cannot contain him? ...
2 Chronicles 2:6
"I am the Alpha and the Omega,
 the first and the last,
 the beginning and the end." 
 Revelation 21:13

These verses show among other things God is not just everywhere(he has to be everywere if the Heavens of Heaven cannot contain Him, also elsewere The whole earth is filled with His glory). It shows He is everytime as well "Before Abraham was I AM" in the context was demonstrating that Jesus is God, but also denoting that He is in a state of presently exsisting (I am) in the past (before Abraham was). And also I AM the Alpha and the Omega, not I Was the alpha and will be the Omega. Not I was the first and will be the last.

Omnipotence: “With man this is impossible, but with God all things are possible.” Mathew 19:26 "I can do all things through Christ who strengthens me." Philippians 4:13 It's pretty clear, is anything left out when He said All things? No.

Omniscence: "Do you not know? Have you not heard? The LORD is the everlasting God, the Creator of the ends of the earth. He will not grow tired or weary, and his understanding no one can fathom" (Isaiah 40:28)

"Nothing in all creation is hidden from God's sight. Everything is uncovered and laid bare before the eyes of him to whom we must give account." Hebrews 4:13

Apart from the other scriptures that talk about His knowledge and Wisdom, if we had none of them we could infer from his attributes of being everywhere at every time and being all powerful that He would have to know everything since He created all things and all things are in Him.

Omnibenovolence:

  "Greater love has no one than this, that one lay down his life for his friends."
John 15:13
"For God so Loved the world, that He gave His only begotten son, that whosoever believes in Him, shall not parish but have everlasting life."
John 3:16
"He that loves not knows not God; for God is love."
1 John 4:8
"Neither is worshiped with men's hands, as though he needed anything,
 seeing he gives to all life, and breath, and all things;"
 Acts 17:25

God is not just all loving He is Love itself.

A google search looking for a meaning/definition of the omni prefix will give you a bunch of references that saying it denotes an "all-" meaning. The suffix words can be defined as:

Potency: power; authority. - Luke 1:37
Presence: the state or fact of being present, as with others or in a place. - Psalm 139:7-12
Science: the state of knowing. - Job 37:16, Psalm 147:5, 1 John 3:20
Benevolent: Characterized by or suggestive of doing good. - Matt 19:17

As those verses describe, God is all-powerful and all-authoritative, all-present (God's presence is in all places at once), all-knowing, and is all(perfectly)-good.

I'm sure there are other references, but for example:

Omnipresence: Psalm 139:7-12 describes God being everywhere.

Omnipotence: I can't think of a Bible verse that says, "God is omnipotent". But Deut 4:37 (and many similar verses) talk about God's "might power". Gen 17:1 and others refer to God as "Almighty". Job 11:7b "Can you find out the limits of the Almighty?"

Omnibenevolence: Luke 18:19 "So Jesus said to him, 'Why do you call Me good? No one is good but One, that is, God.'"

I can't think of a verse for omniscience off the top of my head.

Omniscience: OT reference Psalm 69:5 - if He knows all our sins, He knows it all and NT reference 1st John 3:20 comes right out and says that "...God is greater than our heart, and knows all things."