Several names have been thrown in the hat, but the reality...to this point in time, at least...is that no one seemingly knows, or at least is willing to say definitively, who it is. Since I'm not a finesse fisherman, I didn't even realize until yesterday, while researching a different topic on the Internet, that there was any question about who should get credit for introducing this technique.
Accordingly, don't expect me to throw in my two cents' worth in this forum. Rather, I only will share what information I found online and let y'all make your own decisions.
One theory I found advanced yesterday is that finesse fishing perhaps may have "several fathers." Names mentioned included Charlie Brewer, Don Iovino, Ned Kehde, and Ray Fincke. A fella from Tennessee (who didn't offer his name), however, said he "recognizes Charlie Brewer as the father of finesse fishing, with a nod to Billy Westmoreland."
While there doesn't appear to be any simple way to determine who the godfather of finesse fishing really is, we do know who coined the term "finesse." The earliest reference to using light tackle (spin or casting), specifically for bass, was made in Jason Lucas' book entitled Lucas on Bass Fishing, which was published Jan. 1, 1949. Lucas devoted an entire chapter to the subject. Another very early example, from someone who is considered a bass-fishing pioneer (historically speaking), came from Robert Page Lincoln's 1952 book, Black Bass Fishing, and the chapter titled, "The Gentle Art of Spinning."
Billy Westmoreland's name also comes up in various newspaper articles dating back to 1958, in which the Hoss Fly is mentioned. So he was using light tackle to fish that bait at least that long ago. In fact, some of the articles even contained pictures of large trout being caught on the Hoss Fly, and Billy would later go on to become renowned for his prowess with hair jigs for smallmouth. He certainly could be considered the godfather in that specific sense: hair-jig smallies in clear highland impoundments.
It also was in the 1950s and 1960s that some Ozark greats entered the discussion. Guys like Guido (Hibdon?), Ned Kehde, Ray Fincke, Drew Reese, and Chuck Woods saw the birth of things like the Beetlespin and Puddle Jumper. That time frame, too, is where you can peg the start of maribou jigs, Bass Buster, and Virgil Ward.
In 1966, Bill Binkelman penned what might be considered the first treatise on light-line fishing exclusively, with the publishing of Nightcrawler Secrets. Though it was specifically a live-bait technique, its focus was on using the smallest hooks, the lightest split shot, and 4-to-6-pound line for trophy bass and walleye. He was centered in the upper Midwest, Milwaukee to be exact, and was the guy who created Fishing Facts magazine. In December 1963, when he started the publication, it originally was called Boston Store Fishing News and Wisconsin Spoonplugger.
It was the late 1960s when Charlie Brewer developed Slider fishing. It didn't really catch on until the first articles he was asked to write on the method appeared in Fishing Facts in 1971 and 1972. This is where we also can peg the term "finesse" to bass fishing, as Brewer and another gentleman, Charlie Ritchie of Texas, both wrote articles on Slider fishing for bass and used the term "finesse" in print to describe the technique.
According to Charlie Brewer, Jr., "The name 'finesse fishing' came about because Charlie Sr. was forced to play bridge with his wife of 56 years, and it was a term that applied to their card game. He applied it to fishing, too. The term 'finesse' in bridge goes back to the 1930s, so this makes sense."
This period also is the time that Dick Trask and Don Iovino were doing their light-line specialty fishing out West, known as split-shotting and deep-doodling. However, they reportedly didn't refer to what they were doing as "finesse fishing," at least not quite as early as Charlie.
In summary, there's just no foolproof way to peg a specific person with creating finesse fishing and being the godfather. Charlie Brewer, Sr., however, was the man who first applied the term "finesse" to his Slider bass fishing, and for that reason, gets some folks' nod for the title. As one gent allowed, "It's good enough until when or if someone comes up with something definitive."