Why is the name half-adder used to represent the half-adder?
With a full adder, you can get all outputs from (0,0) to (1,1). With a half adder, you can never get (1,1), which makes it, in a way, incomplete.

Also, in practice a full adder is used more than a bare half adder. For example, addition of binary numbers (one of the most common operations needed by a microprocessor) is done by chaining full adders:



(4-bit adder, taken from Wikipedia)

In the end, since full adders are the ones used for adding, we call them "full" adders.

And since a full adder can be made by joining up two half adders with an OR gate, the half adder becomes "half" an adder.