In the first half of the 19th century, chemists struggled to organize the growing list of elements in a way that could help predict their properties. It wasn't until the late 1860s that Russian chemist Dmitri Mendeleev invented what would afterward become known as the periodic table. The layout of the table has been expanded and altered over time, but the table still fulfills the same fundamental purpose. In the 1950s, American entertainer Tom Lehrer wrote and performed a song that set the periodic table to music. You can still use the same song to help memorize the table.
Find the tune to Gilbert and Sullivan's "I Am the Very Model of a Modern Major General" using the first link under Resources. The link includes the sheet music and a sound file so that you can listen to the music and sing the words.
Refer to the second link under the Resources section, which includes the words to the original Tom Lehrer song. There's also a short clip of the song so that you can sing along to practice the words.
Sing the words to the music of the tune to memorize the names of the elements on the periodic table.
Keep practicing the song until you can remember the words. The secret of all memorization is repetition.
Writing down the names of the elements on the periodic table is another good way to practice memorizing them.