How do I choose which quotations are best to analyse that aren't the same as everyone else?

Choose the quotations which appeal to you the most as they are the ones you will be most passionate to discuss. Try to pick compact but dense quotations that you can analyse in more than one way. If it is to learn for an exam, select versatile quotations that you can analyse with several themes. For example, I am interested in the presentation of nature and the presentation of women in Paradise Lost, so a good quotation for me to learn would be the description that Eve 'rose' to tend to 'her nursery; they at her coming sprung / And touched by her fair tendance gladlier grew'.
If you have a lot of quotations in common with other students, then consider how you are analysing the quotations as another way to stand out from the crowd. For example, many students struggle to analyse the aural qualities of a poem or poetic play (like Shakespeare) so it can be really beneficial if you notice something different about the sound of a line in comparison to the rest of the poem. Or you could make your essay unique by pointing out where the language in the quotation comes up elsewhere in the text (if you're studying Shakespeare, this is made easier with the resources on Open Source Shakespeare).