The roots, when you've got enough of them, can be used to dye with. They'll give you a range of reds. I've used cleavers roots to dye with and gotten a really pretty salmon color on the wool, which I spun into a pretty, salmon-colored yarn.
This little plant has a lot of uses and these are just a few that I've touched upon.
It's easy to mistake Lady's Bedstraw for Cleavers. They look very similar and are related (they are both in the madder family). To tell the difference, all you need to do is touch the plant. If it 'cleaves to you', then it's cleavers. If it's content to let your fingers go without clinging to them, then it's lady's bedstraw.