Memorization (which in this discussion I am using to mean memorization of poetry, especially, but also prose literature) was once a large part of a humane education. The Greeks seem to have been able to recall vast stretches of Homer, and, later on, of Euripides. Aristophanes seems to assume that his audience will get his literary allusions not just to comedy and drama but also to the lyric poets. Plato, not always the most reliable judge, even thought that literacy had the serious drawback of impairing the memory. The Romans, too, memorized their own literature, and to judge from misspelled grafitti, even people of poor education had memorized the Aeneid. Memorization never died out as an educational tool, but it is in the Renaissance that memorization techniques become an obsession--a subject on which Frances Yates has written. I think the reason was the great gap between late Medieval folk culture and the vast amount of classical literature that educated people were supposed to master.
The most obvious advantage of memorization is that it provides a common core of historical and literary knowledge on which every educated person can draw. In this sense, it is important not to memorize only what one happens to like: Better 1000 lines of Homer or Shakespeare than 10,000 lines of John Clare.
A second advantage is that someone with a well-stocked memory has trained his sensibilities on "the best that has been thought and said." It is more difficult to be content with writing garbage, if the writer cannot help quoting and echoing what is well-written. But a store of good poetry is also a resource and comfort in times of trouble. Prisoners deprived of books can only turn to what they have committed to memory. I recall reading the memoirs of Alexander King, a small-time journalist of the 40's and 50's who became a famous curmudgeon on the Jack Paar show. King had been a morphine addict and, when sent to Lexington, had only his memory to rely on--though he remembers once masseur who could quote all of Eliot's Prufrock. The way the world is going, you never know when a few thousand lines of verse may come in handy.
The reading and memorization of poetry is an important part of becoming fully human. All great literatures begin in verse, and we can never divorce ourselves, without peril, from our beginnings. Back in the 1970's Fredrick Turner, a poet and essayist who has appeared several times in Chronicles, co-authored an essay with a neurophysiologist (I think his name was Ernst Poeppel) to show the metred verse had a strong effect on integrating the two halves of the brain--an effect produced by neither prose nor free verse.
Although we are fighting a decidedly uphill battle in trying to get our children to memorize poetry, it is an important fight. We started with Mother Goose and went on to Hilaire Belloc's children's rhymes, then R.L. Stephenson. We also made them memorize some of Macaulay's Lays of Ancient Rome, and, among other things (such as speeches from Shakespeare), "The Assyrian came down like the wolf on the fold."
Poems and songs are an overlapping category, and songs teach children to appreciate form and rhythm without being aware of what they are learning. English and Scottish ballads are also good as are their American descendants. They teach heroism in the face of impossible odds, that even outlaws can have honor and love. (I am very fond of outlaw ballads, not only "Johnnie Armstrong" but also "Streets of Laredo," and even such almost contemporary songs as Woody Guthrie's "Prettyboy Floyd"--admittedly a rather Marxist take on the hero--and Marty Robbins' "El Paso." I put my children to sleep, singing the old songs like "Barbry Allen" and "Frog Went a-Courtin" and even even sentimental parlor songs ("Old Black Joe," "My Grandfather's Clock").
The poems I found unsuccessful for this purpose were either too intellectual, too Romantic in sensibility (Don't try "I fall upon the thorns of life, I bleed.." with a 10-year old boy), or too merely descriptive. In my experience, as both child and parent, children like action, crisis, battle, and death.
If anyone has any specific questions about poems to recommend or advise against, I'd be happy to answer them.