I'd go with a Nubian for a cheap weed eater. Excellent milk, great for cheese or soap. Or, you could go with a Boer; a little more pricey due to the increased demand for meat. You could also get a pygmy; some have excellent dispositions. p.s. They'll also fertilize the rest of the lawn while eating the weeds. Protect the rose bushes though, as goats consider these to be some of the tastiest of all weeds.
Expect to pay about $35ish for a young one... Feed (besides the weeds) will run about $3 a week, making a goat much cheaper to take care of than a dog. I just weeded one of my wife's flower planters with one yesterday. He did an excellent job. Ohhh, and get a doe, not a buck.