It's just a short slang word - not a shortening.
Back in the time before broadband internet, players in groups would communicate via text chat instead of voice chat. It was sometimes necessary to signal that unintended enemies were soon to be or currently engaged.
In Everquest for example, this signal needed to go to:
- the puller, who could either halt pulling or kite/feign/suicide the current pull, sparing the group.
- the tank so they could stop watching for the puller's return and instead engage the local enemy.
- the casters, so they would know to stop meditating (they were blind while meditating).
"Add" was a great term to convey the danger of this situation. In addition to being a meaningful word, "Add" is positioned directly on keys where the player had their fingers. This makes it an easy signal during crunch time - better than the most common alternative: "Pop".
The use of the term "Add" has, over time, come to mean any enemy in a boss encounter which is not the boss. Usually the secondary tank ("off tank") will pick up the adds, or a ranged player will kite them.