How can I remember Juliet's balcony speech?

If you're asking how best to memorize this speech, there are a number of methods that actors use to help them memorize lines. Shakespeare is sometimes difficult for actors in this respect, because the language is so very different from the way we use contemporary speech. This presents a special set of problems, because the imagery and words used are often unfamiliar to actors and do not conform to typical everyday speech. Sometimes focusing on the imagery can be helpful in remembering it.

I had to memorize this speech in college for an acting class, and I recall trying to put myself in the position of a girl who was so infatuated with her new romance that she thought of nothing else. In the speech she is trying to figure out how they can be together, and since it is a soliloquy (revealing her private thoughts, as if she is talking to herself), one way to memorize it is to connect her words to her thought process. As she thinks about Romeo, she thinks of ways to address the obstacles that keep them apart.

For example, when she says the following, she is using logic and common sense to find a solution:

That which we call a rose
By any other word would smell as sweet;
So Romeo would, were he not Romeo call'd,
Retain that dear perfection which he owes
Without that title. Romeo, doff thy name,
and for thy name, which is no part of thee,
Take all myself.

She is suggesting that Romeo should reject his name and birthright, so that they can be together. It is an impossible fantasy, and yet Romeo overhears her and plays along, because he also wants to escape the difficulty surrounding their family situation. This all makes logical sense totem, so might help make their words feel more natural and thus easier to remember.

Another way that works for me in memorizing a long speech is to recite it over and over again, beginning with the first line, and each time going back to the first line and adding one more line each time. This means you'll remember the lines by saying them over and over, and the earlier lines will be rooted in your  memory as you add more on. Good luck!