The Apollo 8 patch is a lovely design. The general shape of the patch reflects the shape of the Apollo command Module, and is roughly the shape of “A” for Apollo. The red “8” denotes both the mission number, and the circumlunar trajectory of the mission.
Jim Lovell recalled:
The design of the Apollo 8 patch was quite unique. Borman and I were in California working on our Apollo spacecraft when we got word that our mission had changed. We were going to take McDivitt’s spacecraft and make a circumlunar flight around the moon.
“On the way back to Houston the next evening, Frank was flying the airplane, and since I had nothing to do, I sort of sketched out what I thought would be an appropriate patch. After I returned to Houston I gave my sketches to the NASA artist who made the final drawing.
In May of 2008, Ed Hengeveld and Noah Bradley revealed information Noah had uncovered about artist Allen Stevens. He was an artist who had created no fewer than four of the crew patches during the early Apollo program. As it turns out, that number was very nearly five. This proposed design by Al Stevens for the Apollo 8 patch had been known for several years before the full story was revealed. This is what I wrote about it before the Al Stevens connection became known:
Here is a photo of a patch reputed to be an early version of the Apollo 8 patch. When I first saw it I thought to myself, ‘Wow, someone has guts to try and palm this off as genuine. Who on earth would believe this story?’ It looked to me as if someone with knowledge of both the final Apollo 8 patch (which has approximately the same shape as the center vignette of this patch) and Lovell's Apollo 13 patch (which features three horses) cobbled together this pastiche. Obviously, this could have occurred no earlier than late 1969, well after the Apollo 8 mission was flown. However, after discussing this image with some other knowledgable people, I’ve come around to a state of ambivalence about it, and offer it here for your consideration.
Here are some details to ponder. First, certain elements in the style of the artwork (see detail photos below) is similar to both the Apollo 1 and Apollo 7 patches — in the rough style of the outer border, and in the ‘scribbling’ style evident in the areas that would normally be flat (i.e. solid) colors — suggesting the possiblility that the same artist may have executed all three. Also, the lettering on three of the four Apollo patches beginning with Apollo 7, was in “Eurostile Extended” — the exception was the final Apollo 8 patch, but this purported “early” design also uses Eurostile Extended.
Second, the legend “Apollo VIII” has clearly been pasted over the original legend. (Again, see the detail image below.) Here is some background information that could account for this:  The crew originally slated for Apollo 8 was Jim McDivitt, Dave Scott and Rusty Schweickart, while the crew slated for Apollo 9 was Frank Borman, Mike Collins and Bill Anders. In July 1968 Mike Collins was diagnosed with a bone spur that required surgery, which resulted in his backup, Jim Lovell, replacing him on the crew (Collins, p.288).  In the original plan of missions, Apollo 8 was to be a test of the lunar module in Earth orbit (Maynard’s “D” mission), and Apollo 9 was was to test the lunar module in cislunar space (Maynard’s “E” mission). When it became apparent that the Lunar Module would not be ready for a flight in 1968, planning was begun to remanifest the missions. While this planning began as early as the spring of 1968, it was not until October that the plan became generally known (Murray and Cox, pp.316-324). When the flights were remanifested, the crews were swapped. So, there was a window of a few months when the crew of Borman, Lovell and Anders were scheduled to fly the Apollo 9 mission. If this patch were created during that period, the original legend at the bottom would have read “Apollo IX”; and when the crew swap occurred, the new legend may have been pasted over the original.
Third, and to me the most significant, this photo actually has an official NASA photo ID — S68-53480.
The problem was ... I had it completely backwards! In the light of the information uncovered by Noah Bradley, it now seems clear that Al Stevens had executed this version for the Apollo 9 crew, after Collins had been replaced by Lovell. What happened after that is not known for sure, but here’s my conjecture: the crew had accepted the patch, but then the Apollo 8/9 mission swap occurred, and the patch was altered to reflect the change in mission number (not by Stevens, though: he was very precise with his lettering). Then it was sent to the MSC Public Affairs Office, which proceeded to photograph it in preparation for a press release. When it became clear that Apollo 8 was going around the moon, Lovell appreciated that the number “8” perfectly reflected their new mission, and came up with the idea of an “8” looping around the Earth and Moon — which fit nicely into the CM shape from Stevens’ design. The elegance of this concept was compelling, and the crew adopted the new design. A year or so later, when Lovell was looking for a patch design for his Apollo 13 command, the memory of the three horses from Stevens’ design must have influenced him when he saw the Lumen Winter mural.
So, even though Stevens’ design for Apollo 8 wasn’t used, elements from it clearly influenced the final design of two Apollo patches.