It may be hard to find, but I recommend a Honda C50. They are very reliable. They use about half the feul most other mopeds do (around 140 MpG, sometimes they use even less). It's a geared moped with power throughout the entire revving range, that gives a very comfortable drive. 100.000 miles on just one cylinder and piston is no exception, if you treat it good it should be able to do more than that. It's got legshields, footpegs, a buddyseat (seat for 2), winker bulbs, good lights, brake lights and it's an absolute classic. Best of all is that it's got a fourstroke engine, so no oilmixture but leaded gas and no stinking blue fumes from the exhaust while the exhaust is clogging up with black goo. It has a really cool engine sound (espacially when you remove the muffler!). The Honda C50 is the best sold moped in the world (just not in the US). There were more than 30 million peds sold wich says enough about what other people think about that bike.
And besides all that it's really easy to make this bike run a lot faster if you've got the version with contactbreaker-ignition, it doesn't work on the one with CDI ignition. Inside the flywheel there's two little swingarms that will move outwards when the revs rise because of centrifugal forces. When they move outwards they shift the contactbreaker-cam so the ignition will be retarded and the bike just doesn't have power at high revs. This is how the bike is limited to 30 Mph. If you block the swingarms (with a little piece of iron wire or two parker-screws) the ignition wont be retarded. If you combine this with setting the contact breakers to start opening about half an inch before the F-mark on the flywheel reaches the top mark in the carter (advancing the spark a bit), than this ped will run about 40-45 Mph.
P.S. if the cylinderhead is making a loud ratling sound then it's not broken, you just have to reset the valve tappets. If you set them right (0.05 mm gap) you should still be able to hear the tappets, but it shouldn't be very loud.