Wring means to twist, squeeze, or clasp firmly, especially to extract liquid. It is the correct word in wring [one's] neck, meaning to choke. Ring would almost make sense there, as wringing a neck involves holding one’s fingers in a ringlike position. Nonetheless, wring is conventionally used in the phrase.
Examples
Ring is commonly misused in place of wring in the phrase wring [one's] neck—for example:
Sure, your cast mates will ring your neck, but at least you’ve escaped a horrible scene. [Chicago Tribune]
it’s just incredible Sabu was able to deal with Heyman for so long without ringing his neck. [Pro Wrestling Torch]
These writers get it right:
Most people raise a rooster until he is six or seven months old, then they take him out behind the house to wring his neck or take a hatchet and cut his head off. [Wetumpka Herald]
They will wring their necks or stuff them still alive into barrels to die from suffocation. [The Moderate Voice]

