Why are plant buds called 'eyes'?
Just take a look at a potato and observe each of its eyes. You will see that each of them has two parts:

1) An "eyebrow" : this is a vestigial leaf

2) A small bud in the axil of that leaf 

Thus, the two of them together form an eye + eyebrow

The eyes are not evenly distributed in a potato plant. One side may have just one or two but the other side will have lots of them. This is because that is the growing spot!

Examples: