Jeremy Walker at Reformation 21 has a very thoughtful blog post on the importance of Bible memorization. His key point is that:
One of the consequences of the internet-trained brain seems to be an inability to hide very much - not much of the Word of God, to be sure - in our hearts. That results in a crippling weakness in the battle for godliness.
If we are to be holy we need to hide the word in our hearts, and that means a deliberate commitment to memorisation and meditation. It means a refusal to allow our brains to be trained by the world, a resistance to the laziness that the interweb can breed in our all-too-susceptible minds; it means a commitment to holiness that is willing to re-train and develop the faculties of our hearts contrary to the trend and tendency of the age in which we live, and to make sure that we pack into the armoury that array of weaponry necessary for the constant fight against ungodliness, temptations within and without.
It's definitely worth taking the time to read the whole post.