Our Lady's Psalter
This is a good question, and it's a piece of rosary history that I don't think I've mentioned here. Here's a short review.
Priests, monks, nuns and other full-time religious people in the Catholic Church have "official" prayers to recite every day. These prayers used to be referred to as the "Divine Office" and today are called the "Liturgy of the Hours." These are seen as prayers offered on behalf of the whole church, for the entire world. The Divine Office is based on the Psalms, together with other prayers, hymns and readings, and those who pray it actually do recite the 150 psalms at least every week. (If I recall correctly, in the Middle Ages the Office was heavier on the Psalms and lighter on the other stuff.)
Lay people wanted to participate in this "official" prayer of the Church, too, including those who couldn't afford an expensive psalm or Office book and those who couldn't read. While I'm sure there were people who did memorize all 150 psalms, it was much more common for such people to simply recite the Our Father (Pater Noster) 150 times, once for each of the 150 psalms. This is what the paternoster beads were used for -- to keep count. We have records of practices like this several centuries earlier than the "rosary" as we know it today.
Later elaborations included adding a "Hail Mary" after each "Our Father," or just saying 150 "Hail Marys", and this was referred to as reciting "Our Lady's Psalter." To this day, you are likely to hear a 5-decade rosary or a loop of 50 beads (or 50 plus markers) referred to as a "psalter" in German, to differentiate it from other forms of beads (such as strings of 10, or 6- or 7-decade forms).
Other devotions also used the same beads. Various people wrote devotions for the beads, including sets of 150 rhymed verses, one to add to each "Hail Mary" of the 150 (which again required a book). One of the early rosary manuals has been republished in recent years and has been surprisingly popular because it provides 150 Scripture verses -- again, one for each bead. The modern, newly re-invented devotion is called a "Scriptural Rosary," and has many enthusiasts.