The Gemini 8 crew patch can be clearly seen in these closeups from preflight portrait photo S66-27513.
This patch may have been produced in extremely limited quantities[1].
Most souvenir versions of the patch overlook the fact that the symbols at the bottom are not "IIVIII" but the Gemini symbol followed by "VIII" for the mission number. The left and right upright parts of the Gemini symbol curve inwards, as can be seen on the crew patch.
A high quality modern replica of the Gemini 8 crew patch, produced by AB Emblem in 2010 in collaboration with Gene Dorr.
A fair replica of the Crew Patch, but the left two characters towards the bottom are shown as Roman numerals rather than the in-curving Gemini symbol of the original.
The AB Emblem souvenir patch for the mission has no outlining around the Roman numerals and uses a straight "II" instead of the in-curving Gemini symbol for the left two characters.
The sole example I've seen to-date of this patch from an unknown manufacturer appeared in a collection of vintage Gemini crew patches. It seems likely the patch was produced by the manufacturer of some of the recognized Gemini crew and crew souvernir patches, but perhaps didn't get beyond the sample stage.
[1] Donnis Willis has reported that Dave Scott told him only four examples of this particular patch were produced.
[2] Heritage Auctions September 21, 2007 lot 25074 at $7170 (inc buyer's fees) [FLOWN]
This example was given to Joe Garino by Dave Scott after the flight, and Garino stated that it was flown.
The patch looks to neat to have been sewn to a suit, which would make it a flown souvenir.