When I first got serious about memorizing and meditating on God's Word, I started by making a list of verses that were especially liked or were helpful to me as I came upon them. Then, one at a time, I would memorize them (as I kept adding to the list).
This month I have tried a couple new things. One was memorizing 10 or so verses on a certain topic. I used a book called Light for My Path (which has verses organized by subject), and this month I memorized verses on brotherly love.
Another thing I started which I have really enjoyed is doing one chapter per month. For December I did (or am almost done with) Psalm 18. I must admit, lots of days it has been a real challenge to commit time to memorizing when I don't feel like it. God has really helped me though and over the past many months has been strengthening my memory (it really is a muscle!). I have tried to devote about 10-15 minutes from my quiet time in the morning plus about 10 minutes before going to sleep to memorizing... longer than I had originally expected, but it's worth it. :)
I think I'm going to do Psalm 103 for January. It is such a wonderful reminder of the great things God has done.
Praise the Lord, O my soul, and forget not all His benefits--
who forgives all your sins
and heals all your diseases,
who redeems your life from the pit
and crowns you with love and compassion,
who satisfies your desires with good things
so that your youth is renewed like the eagle's.
Would you like to share any memorizing techniques? :)