A very simple and cheap robot arm I have found is the MeArm. You can find a wood version on Amazon for about thirty or forty bucks, or, if you have a 3D printer, find the files online and make one yourself. There are other similar arms out there for relatively good prices, my favorite being the OWI Arm Edge (although it is anything but precise).
You can just Google "simple robot arm" and as always with Google, about a thousand semi-relevant results will come up. One of them is sure to fit your needs.
I found several great arms by searching "simple Arduino projects" and scrolling through a few of the pages that popped up. Some of them are 3D printable, some you have to buy as a kit, and some come pre-assembled. It all depends on what you are looking for, and how much you are willing to pay for it.
If you have a really fat wallet, I would suggest you look up the U Arm Swift Pro, which has almost no limit to it's capabilities. 3D printing, laser engraving, facial recognition, teachable movement; you name it, this arm is probably able to do it.
Have fun looking trough the endless lists of robot arms that Google is sure to find for you. There are also several videos on YouTube with "Top 5" lists of robot arm kits and how they work.