You might want to consider building, and designing (or following a standard design for) a robot arm, using "servo backets"1, which are designed to be used with standard servos (i.e., the MG995 or MG996R). Here is a common example of one such robot arm:

These brackets are cheap and widely available, as a quick search for servo brackets, or robot brackets will show you.
There are many types/forms/shapes available, and they are all compatible with each other:
- Short U bracket;
- Long U bracket;
- Oblique U bracket;
- I bracket;
- L bracket
- Multi Function bracket;
- Body Plates;
- Elongated, or Large, U bracket;
- and others.
Note that these names are not standard, merely descriptive, and may vary from supplier to supplier. Here is a Short U and a Multi Function bracket

These brackets are reusable, so, once you have completed your robot arm, should you become tired of it, you could dismantle it, purchase some additional brackets and make a humanoid (upright) robot,
or a hexapod,

or whatever.
As the first photo (of the robot arm showed), a simple robot grabber/gripper is also available for the end of the arm:

These grippers, and the brackets themselves, are widely available on eBay. Note that it is invariably cheaper to buy the brackets independently, than the whole arm kits that you see, by about 50% usually. Although having said that, the 21 DOF humanoid robot kits are actually the cheapest option, if you can get one for £45, and you think that you will need all of the extra brackets at some point in the future. You will also require bearings (and cups), so make sure that any kit that you buy has those included.
The parts that you will require for the robot arm are as follows:
- Multifunctional bracket x 5
- Long U shape bracket x 3
- L shape bracket x 1 (to attach the claw)
- Acrylic board x 1
- Cup bearing x 4
- Screw nut x September 1
- Alloy mechanical claw x 1
- U beam bracket x 2 (for the base)
- MG996R x 6
- Servo Extension cord x 3
- Servo horns x 6
Here is a better view:

1 There is a correct name for them, other than servo brackets, but the real name escapes me.