They're known by dozens of common names, including clivers, goosegrass, stickywilly, Stickyweed, catchweed, and bedstraw (lots of plants are called bedstraw). What did you call them when you were kids? We called them "stickybuds", but we made that up ourselves and I don't know anyone else who calls them that. Maybe that's how they acquired so many names - from children.
You can get some thick patches of cleavers in new or neglected ground, but they're very easy to dig out. You don't even need a spade, you can just pull them out with your bare hands. The roots are very shallow and not extensive. Their big propagational trick is the sticky seeds, which stick to birds' feathers and animals' fur just as easily as dad's jumper, and get everywhere. They'll also be in any consignment of fresh manure you bring to your plot (I'm told horses are very fond of eating cleavers, and I know my chickens love them), and probably in any compost you use as well. Just hoik the seedlings out when you see them. Cleavers are no big deal. Not like couch grass, bindweed or ground elder. Those are the weeds you lose sleep over.
According to http://www.herbalremedies.com/:
The dried or fresh herb is said to have anti-tumor, anti-inflammatory, astringent, diaphoretic, stimulant, laxative and diuretic properties.
I've read in several different places that the seeds can be roasted and ground, and used like coffee, but I've never tried it. And The Really Wild Food Guide has two recipes for the leaves (not the stems, though, which are very tough): Spicy Chicken and Goosegrass, and Birch Sap and Cleavers Risotto.
Cleavers get my vote. They're not much trouble in the garden, and they gave me so much mischievous fun as a child. I'd have to say that cleavers have probably enriched my life as much as any vegetable. They're my favourite weed.