How would you biblically show someone that God has given us free will because he loves us?
Of course, the expression "free will" is not found in the Bible, but the idea that God has given us a choice and that we are therefore responsible for our decision about what to do with that choice is found throughout the Bible. It is almost as though it is so biblically obvious that specific verses are not used. This is a basic underlying assumption of the Bible.
However, you ask for passages, so I will give you some. I believe Deuteronomy Ch 28-30 is a good example. God presents to his people a choice to obey or to rebel. He lists a whole list of blessings for choosing to obey (Deut 28:1-14) and curses for choosing not to obey (The rest of Deut 28). The culmination of the incredibly emotional appeal is found in Deuteronomy 30:19-20 summarizes the thought. "This day I call heaven and earth as witnesses against you that I have set before you life and death, blessings and curses. Now choose life, so that you and your children may live and that you may love the Lord your God, listen to his voice, and hold fast to him. For the Lord is your life, and he will give you many years in the land he swore to give to your fathers, Abraham, Isaac, and Jacob." It is hard to miss the call to choice, as well as the consequences of our choice here. The basis of the entire Deut 28-30 is God’s love for us (as well as his justice, of course). Then there is the classic in Joshua 24:15 where Joshua calls Israel to "choose today who you will serve." Also noteworthy is 1 Kings 18 where Elijah demands that the people choose between Jehovah and Baal. I would add Ezekiel 18 and Ezekiel 31 where God tells us that it is our choice to sin which determines whether we will die spiritually or not.
I believe that even if we did not have hundreds of passages which prove that mankind is given freedom to choose to do good or evil, the mere fact that "God is love" demands that God give us a choice. Love, by definition, cannot be ordered or forced. Love is a choice. The command to love God with all our heart, mind, soul and strength implies that we are free and able to do so. Jesus’ command to give up everything and follow him (Luke 14) imlies that to become a disciple involves a free choice. Otherwise his commandment is nonsense. I hope this will get you started.