In one sense, "effective" prayers mean that your prayers are answered in the way you desire them to be answered. So, for our prayers to be effective in that way, they must line up with God's will (1 John 5:14–15). We know God's will is revealed in the Bible, so praying Scripture is effective in this sense.
The Bible tells us that God hears our prayers and answers them. It also tells us that our access to God comes when we accept the sacrificial death of His Son, Jesus Christ, as our substitutionary punishment for our own sins (Hebrews 10:19–25). We must believe that Jesus then rose from the grave, having power over even death (Romans 10:9). Salvation is from His grace and received by faith in Him (Ephesians 2:8-9).
Believers are given the indwelling Holy Spirit, and He works to transform us and make us more like God (1 Corinthians 5:17). While salvation is not about our works, God does call us to a life of holiness (Ephesians 2:10; 1 Peter 1:14–16). God tells us to strive after righteousness, and that righteousness is a foundation to effective prayer (James 5:16; 1 Peter 3:12; Psalm 34:15; Proverbs 15:8).
We are also commanded to pray continuously and with thanksgiving (1 Thessalonians 5:16–18). We should pray about all things (Philippians 4:4–7), and our prayers should be focused on God's glory, not our own (James 4:2–3).
Following these guidelines is by no means a formula. God does not want us to know a formula, but to know Him. The primary purpose of prayer is not getting the things we want, but fellowshipping with God as Father and exercising the privilege He's given us. Prayer is effective in accomplishing kingdom work and God invites our personal requests as well (Matthew 6:9–13; 7:7–11; Luke 11:2–13), but He is not a cosmic genie we coerce into granting our requests by praying a magic formula.
Praying Scripture can be a very effective way to pray in line with the will of God. Being a student of the Word of God is the best way to know God. Scripture reveals truths about God's character as well as His desires and plans for us and our worlds.
Scripture is also filled with examples of how to express ourselves to God. The Psalms are essentially prayers. Jesus' longest recorded prayer is in John 17. Many of Paul's letters include prayers. The more we use and know the Bible, the more we can use it to express ourselves in prayer.
Though we can use Scriptures to form our prayers, be aware that not every experience of someone in the Bible can be transferred to our lives. God delivered Daniel from death, but not Stephen. Properly understanding Scripture in its context is key to applying Scripture to our prayers.
While we can and should certainly use Scripture to inform our prayers, remember that in prayer we can also simply pour out our hearts to God (Psalm 62:8). The bottom line here is that when we are honest with God and pray according to His will, our prayers are effective. We know His will best through His Word.