Interesting idea, Mare.
The way I would do it is with a powerchair motor that uses the gearbox shaft as the axle. They usually run on 24V and are max of about 4MPH. Brushed motor, so dead simple to control. Powerful enough to move at least a couple hundred pounds even with just one. Some can do much more than that.
There are two common sizes of wheel for them; one about 14", one about 10". Usually the hubs are separate from the wheel, and the bore hole is different for them. Get a motor that usually uses a 14" wheel, and put the 10" on it instead, and you have even more torque than before, and less speed. Perfect for a wheelbarrow.
I don't think I'll ever build an electric wheelbarrow, but I do have a little "tractor" in the works, to move a small car back and forth in a friend's yard. Once I get past the basic concept stage I'll probably have a thread about it.