I have a relative who has been grilling me about some verses and brought up John 14:14, which he says can't be true since we clearly don't get everything we ask for from God. Do you have any insight?

Do I have any insight? I have an opinion about everything. In context let's start in verse 12, John 14:12-14 “Truly, truly, I say to you, whoever believes in me will also do the works that I do; and greater works than these will he do, because I am going to the Father. Whatever you ask in my name, this I will do, that the Father may be glorified in the Son. If you ask me anything in my name, I will do it."
 
This is one of those sections of scripture that is constantly taken out of the context of what surrounds it by people who believe that God is a genie in a bottle whose every desire is to give us what we think we want. Notice the verse starts out speaking of Jesus' work in the world and us being a part of it. Jesus' work continues, and grows even greater, because of the sheer amount of people involved in His work. This is why today the Gospel is not confined simply to a bit of real-estate that we call Israel (or even the Middle East). Because of what Jesus has done, and is doing in His people, good is happening all around the world. As the Gospel Transformation Bible says "our work for Jesus is really our work with Jesus." 
 
When Jesus says, "ask for whatever you want in my name," He is not saying, "Want a car? Say 'Jesus name.' Want a smaller waist line? Say 'Jesus name.'  Want to win the lottery? Say 'Jesus name.'" There are some very off base people today who tell you exactly that, but instead of whacking God like a piñata, I think we ought to whack them around a bit (with good theology).
 
In the context of the verses, the types of prayers that Jesus is referring to are those that further His work and bring glory to the Father. Most of our prayers today are about our own self-glorification (God give me a man, give me a woman, give me a house, get me a job, give me straighter teeth, and/or make me smarter). Jesus says that the prayers, in this context, that He answers are ones that honor God. We are to make it about His glory, His honor, and His gospel.
 
What should we pray for? How about:

• Salvation for those who need Jesus.
• Courage that we wouldn't be timid about the Gospel but speak with and LIVE with boldness.
• Healing for those who are hurt.
• The health of His church.
• Opportunities to serve one another.

It seems as if almost no one prays, "God, give me opportunity to serve others."
 
Prayer is about changing the heart of God's people, not changing God. When you begin to pray for others your heart towards them will melt…I think this is why we are told to love and pray for our enemies (Matthew 5:44).
 
Again, in context, Jesus follows verse 14 by saying John 14:15 “If you love me, you will keep my commandments." It sounds to me like He is reinforcing that we should be a people who are about His work in the world, not our own selfish wants and desires. Love for Jesus has everything to do with what we pray about, how we live, and how we partner with Him in the world.