As you probably know, snipers are highly skilled and trained marksmen who maintain close contact with enemies and can target and attack them from concealed or distant positions without being detected.
The name Sniper, then, makes a lot of sense when you consider where it came from. In the 1770s, soldiers in British India coined the verb 'to snipe,' meaning a soldier was good enough of a shooter to kill an elusive bird called a Snipe. The verb became a noun by 1824, when the first use of 'sniper' to mean a sharpshooter was recorded.
Interestingly, the name of these marksmen could have just as easily been Skirmisher. It was the popular term in the United States during the US Civil War. However, the term fell out of use (at least for this meaning) and sniper was adopted.