The first listing of reign, according to Merriam-Webster, is a noun from the Middle English word regne. Reign dates back to the 13th century, meaning “a royal authority: sovereignty.” It is mistakenly used in the phrase free reign, which is meant to convey a lack of limitation and not the opportunity to rule freely as a monarch.
Rein, upon second listing in Merriam-Webster, is a noun meaning “a restraining influence.” When correctly used in the common saying free rein, it’s a noun defined as “unrestricted liberty of action or decision.”
Example: I’ve been given free rein to plan the banquet for the fundraiser.
The expression originated in the 17th century and refers to the act of holding onto the reins (i.e., “a strap fastened to a bit by which a rider or driver controls an animal”) of a horse loosely to allow it freer movement at its desired pace.
"The tongues of Angels are not able to expresse what benefits doe redound unto man by the right ordering of the tongue, and what harmes and inconveniences againe, when we give it free reines to lash out." —Alexander Read, The Chirurgicall Lectures of Tumors and Ulcers, 1635
Sources:
“‘Free Rein’ or ‘Free Reign’?” Merriam-Webster, Merriam-Webster, Incorporated, accessed March 19, 2019, https://www.merriam-webster.com/words-at-play/usage-free-rein-vs-free-reign.
Webster's Third New International Dictionary, Unabridged, s.v. “free rein,” accessed March 19, 2019, http://unabridged.merriam-webster.com.
Webster's Third New International Dictionary, Unabridged, s.v. “reign,” accessed March 19, 2019, http://unabridged.merriam-webster.com.
Webster's Third New International Dictionary, Unabridged, s.v. “rein,” accessed March 19, 2019, http://unabridged.merriam-webster.com.