Give up, abandon, as in He laid aside all hopes of winning first prize. [First half of 1400s]
Also, lay away; lay by. Set apart for a reason, save for the future, as in They lay aside enough to pay the rent, or Because coffee prices were rising, she laid by enough for a month, or The store laid away the winter coat I wanted. [Early 1700s] The first variant gave rise, about 1970, to the term layaway plan, in which merchandise is laid away for a buyer who pays a deposit and receives it when payment is made in full. Also see lay down, def. 4; set aside.