The history of the Gibson SG traces back to not only the Les Paul which it replaced, but possibly also a lesser-known model called the "Double 12", which later became the EDS-1275. In late 1957, as Gibson was experimenting with radical new ideas like the Explorer and Flying V, they also began producing a double-neck semi-hollow guitar (albeit in very limited numbers through custom order). This was the first Gibson with double Florentine cutaways. The Florentine cutaway is essentially any cutaway whose horn comes to a sharp point. This style had been around long before 1957, but rarely as a double cutaway. The result was a shape that is similar, but not identical, to what would become the redesigned Les Paul in 1961 (The biggest difference is that the SG's horns are offset). Whether this definitively influenced the SG shape or was just a coincidence cannot be said for certain, but it's a possibility.
The genesis of the SG also traces back to the rivalry between Gibson and Fender. Engineer, and then-president of Gibson, Ted McCarty explained to Vintage Guitar magazine; “Fender was talking about how Gibson was a bunch of old fuddie-duddies, and when I heard that through the grapevine, I was a little peeved. So I said, ‘Let’s shake ’em up.’ I wanted to come up with some guitar shapes that were different from anything else.” This resulted in the 1957 Explorer, Flying V, and Moderne. Upon their introduction at NAMM, the designs achieved their goal by shocking the guitar world. However, in production they proved very difficult to actually sell, because they were seen largely as novelties. Indeed, they proved well ahead of their time, as they would go on to become legendary designs. To make things worse, following the tepid response to the Explorer and Flying V, the Les Paul's sales were proving underwhelming and the Fender Stratocaster was dominating the electric solidbody guitar market.
So, beginning in 1960, Gibson's team of engineers and designers, lead by Ted McCarty and Larry Allers, undertook a redesigning of the Les Paul. While no one person has been accredited entirely with the design, it is believed that Larry Allers had the biggest hand in the SG's design. Clearly, based on the result, this was not just a reworking of the existing Les Paul, but a clean sheet design. The influence of the Stratocaster can be found in the offset double cutaway shape, the thinner and lighter body, the contoured/beveled edges and the offering of a tremolo system. There is also influence from the earlier Les Paul Special and Junior with their double cutaway design, high fret access, thinner body and Cherry finish. In fact, the "SG" name was first used on those two models in Gibson's 1960 catalog as an ordering code (meaning "Solid Guitar"). This became the de-facto name of the new design once Les Paul's endorsement ended.
The body thickness was a drastic change, from 2 3/8" on the Les Paul to 1 3/8" on the SG. That also made it thinner than both the Stratocaster and the Les Paul Special. Another dramatic change was the fret access; the neck on a Les Paul meets the body at the 16/17th fret, whereas the SG's neck is set in at the 22nd fret, the last fret on the neck. While many people believe this makes the neck joint less sturdy, and there is a widespread belief that neck breaks are common on SGs, this is often based in assumptions and anecdotes. As you can see in the picture to the right, the tenon on an SG spans the full width of the neck and reaches about as far into the body as a Les Paul's neck does. Also, mid-way through 1961, a small overhanging lip from the body was added to support the neck. While this may seem inconsequential because of what a thin lip it is, it's actually quite clever because the strongest part of a joint is in fact the glue, not the wood. So for what little is added in bulk, you get a lot of extra strength from the comparatively large surface area of glue. However, the heel would continuously change several times throughout the decade as Gibson sought to make the joint stronger, whether due to genuine issues or simply public perception.
The first SG/Les Pauls were built in late 1960. In fact, some of the first SG/Les Pauls were likely built right alongside older style sunburst Les Pauls, as there is significant overlap in serial numbers.
The brighter tone of an SG, compared to the Les Paul, is largely, if not entirely, a result of the fact that the pickups are placed closer to the bridge than they are on a Les Paul. This could have been a deliberate attempt to compete with the Stratocaster's brighter tone, which cuts through the mix better in a live setting.
Their work quickly paid off, as while Gibson only sold around 1,700 Les Pauls from 1958-1960, it shipped 2,175 SG/Les Paul Standards and Customs in 1961 alone.
In 1963, Les Paul's name was taken off of the guitar and it was renamed "SG" as we know it today. This came from Gibson's internal designation for the model, simply meaning "Solid Guitar". There are various explanations floating around for why this happened. Les Paul himself has given changing accounts over the years and is known for embellishment, making much of his statements unreliable. The most likely story is simply that Les and his wife were going through a divorce at the time and he wanted to divest his name of as much equity as possible, so he let the contract run out, and did not renew it until the divorce was finalized in 1968.
Since then, the SG has remained in continuous production to this day.
For information on the changes made throughout the SG's lifespan, see here.