Q: Are the words "free will" or "freedom of choice" used in the Bible? Even if it doesn't, does it allude to it anywhere?

These exact phrases are not found in the Bible. However, Scripture does talk about the concept at length. After all, free will is the ability for a person to make choices that determine his actions. Below are some verses which strongly show that mankind has the responsibility to exercise their free will and is commanded by God to do so.

Deuteronomy 30:15-19: "See, I have set before you today life and good, death and evil, in that I command you today to love the Lord your God, to walk in His ways, and to keep His commandments, His statutes, and His judgments, that you may live and multiply; and the Lord your God will bless you in the land which you go to possess. But if your heart turns away so that you do not hear, and are drawn away, and worship other gods and serve them, I announce to you today that you shall surely perish; you shall not prolong [your] days in the land which you cross over the Jordan to go in and possess. I call heaven and earth as witnesses today against you, [that] I have set before you life and death, blessing and cursing; therefore choose life, that both you and your descendants may live."

John 14:15: "If you love me, keep my commandments."

Romans 2:10: "but glory, honor, and peace to everyone who works what is good, to the Jew first and also to the Greek."

1 Timothy 6:12-14: "Fight the good fight of faith, lay hold on eternal life, to which you were also called and have confessed the good confession in the presence of many witnesses. I urge you in the sight of God who gives life to all things, and [before] Christ Jesus who witnessed the good confession before Pontius Pilate, that you keep [this] commandment without spot, blameless until our Lord Jesus Christ's appearing."

2 Timothy 2:21 "Therefore if anyone cleanses himself from the latter, he will be a vessel for honor, sanctified and useful for the Master, prepared for every good work."

It is clear that God designed us and calls us to exercise our free will in joyful obedience to him. A number of scholars have written about how this actually works. Dallas Willard, for instance, breaks down a person into "elements." Think about an onion with many layers, with the spirit being the innermost layer and the social relationships a person has as the outermost layer:

         Elements of a Person: spirit --> mind --> body --> social

For Willard, the Spirit of a person is the inner core of being from which actions flow (this is what many would consider “the will”). Our spirit is the power to do what is good or evil. It’s the center of who we are and ultimately who we become. According to the Scriptures, this is what is enslaved to sin (Rom. 6:14-20), is desperately sick (Jer. 17:9), is full of evil (Mark 7:21-23), loves darkness rather than light (John 3:19), does not seek for God (Rom. 3:10-12), and cannot understand spiritual things (1 Cor. 2:14). No matter how hard a person tries, she cannot change this state through a focus on external behavior (e.g. changing the way you dress/behave/talk; attending all sorts of programs, etc.). It is because the will is bent "away from God" and only "toward sin." Summarily, the will is free, but only free toward sin. If we put our faith and trust into Jesus (& his once-and-for-all sacrifice on the cross for our sin, in our place), then Scripture declares that we are given a new spiritual heart that is now responsive, alive and warm to God. This is exactly what the prophet Ezekiel predicted thousands of years ago, before the coming of Jesus:

"I will give you a new heart, and a new spirit I will put within you. And I will remove the heart of stone from your flesh and give you a heart of flesh. And I will put my Spirit within you, and cause you to walk in my statutes and be careful to obey my rules" (Ezek. 36:26-27).

To be sure, our will before faith in Christ was like "stone" (i.e. completely unresponsive, dead toward God). Consequently, we were "free" only to choose sin. When The Holy Spirit comes into our hearts, we are given a new spirit (heart, will) that has the POWER to fight temptation and thus sin. 

Think of it this way: You are confronted with a temptation to sin. Let's say your friends are talking badly about another friend behind her back. That’s when your spirit comes into play. In order for you to commit sin, your spirit ("will") must say “yes” to that sin. If there is no inner yes, there is no sin. Even if you think about sin, that’s not a sin (that’s not even temptation!). Remember, temptation is the thought plus the inclination to sin. It's when you linger over the thought. Sin is when we inwardly say yes to the temptation; when we decide to do it. 

Here's the point: For those that place their hope and trust in Christ, God transforms our spirit to give us the new power to fight against the "inner yes" to sin. So in this way, our free will becomes "free" to follow God in humble and joyful obedience, and "free" from sin.

I know there are many, many more issues concerning free will, particularly as it relates to the foreknowledge of God (e.g. If God knows our free will choices, do we still have free will?). See CARM.ORG's articles HERE.

[Answered by Pastor HM, who is glad he freely choose ice-cream]