An easy way to memorize the names of the 27 books of the New Testament is to break them down into four groups and learn each section separately. Putting the lists to music or a chant and reciting them can help, as can playing mnemonics games.
There are different groupings to break the books down into sections to memorize. A possible arrangement is the life of Jesus and the early church: Matthew, Mark, Luke, John, and Acts; Paul's letters: Romans, 1 Corinthians and 2 Corinthians, Galatians, Ephesians, Philippians, Colossians, 1 Thessalonians and 2 Thessalonians, 1 Timothy and 2 Timothy, Titus and Philemon; the Apostles: Hebrews, James, 1 Peter and 2 Peter, 1-3 John and Jude; and Revelation as its own singular entity. Once you have successfully committed the first group to memory, start studying the second, then the third and fourth.
To help memorize the two- and three- part books, read them as "first" and "second" rather than "one" and "two," for example, "First and Second Corinthians." Using mnemonic devices can also assist with memorization. Make a set of flashcards to practice and play memory games, such as assembling the cards in order face-down and then going down the row, turning each card over after reciting the book. When you miss one, turn all the cards back over and start from the top again. Repeated drills assist in committing the list to memory.