Mystery "organ" on the underside of chicken thighs?

As discussed in the comments under the question, I believe the organ in question is a kidney, from the pocket in the pelvis of a thigh butchered in a fairly unusual manner, with that part of the pelvis still attached.

See page 3.21 in the University of Kentucky's PDF of Chapter 3 of Chicken Anatomy and Physiology. It shows where the kidney's are in the chicken (moderately graphic), and the shape looks quite similar to the mystery item in the original question's photographs, allowing for shrinkage from cooking.

There are no organs on the underside of a chicken, the oyster is simply a bite-size piece of muscle which is tender and usually the tastiest piece of the whole bird. Two of links are about the oyster, which isn't gizzard-like at all. The first link is asking what the livery tasting stuff that sometimes comes attached to chicken thighs is, which is in fact liver left by poorly executed preparation by the packaging company.

Those are the kidneys. I just finished butchering eight chickens yesterday. They look exactly like a little kidney bean (lol). I don't believe they will hurt you, after all people eat beef kidneys, kidney pie, etc. kidneys are usually removed along with everything else. They can be easily popped out with your finger.

The organ meat inside the bony part of a chicken thigh is the kidney. A good cook removes it before preparation; I have never seen it removed by a butcher.

As for the oysters, those are the two "backstrap" or "tenderloin" muscles in the small of the back. They're not organ meat -- just very tasty chicken.

http://www2.ca.uky.edu/poultryprofitability/Production_manual/Chapter3_Anatomy_and_Physiology/Chapter3_excretory.html

Check this from the Kentucky College of Agriculture. Looks like the mystery part is the kidneys. It also says that, "The kidneys are normally left in when a broiler carcass is processed."

Here in Sydney Australia it's always there, always. Fresh chook(whole or thighs), charcoal chicken(bbq), KFC(Kentucky fried chicken), whatever.
It's definitely not the oyster. I thought they might be testicles, but they would only appear in roosters, not chickens. I also thought they might be a kind of bone marrow deposit/reservoir like what humans have in our hips(the putty like stuff they do bone grafts with when you badly break a bone).
IMHO, whatever it is, it's the best bit of the bird there is, YUM!!!

Kidneys are generally found in any chicken thigh portion. I've eaten chicken from many places and never had difficulty finding the kidneys. If you're actually cutting the kidney section out of the thigh, then you aren't leaving the whole thigh.

From the looks of the photos what you're seeing is part of the "oyster". It can sometimes be darker in colour. It's the muscle found near the base of the back where the thigh meets the body.

The texture is slightly different than other parts of the bird. When cooking a whole bird you pop that bit of meat it's concave bone area and it resembles an oyster in shape.

I provided a link to a webpage that has some nice pics of a bird being cut up. You'll see about 3/4 down tha page they have the body split down the breast bone showing the inside along the back bone without legs attached. Notice no organs attached anywhere. I've NEVER seen any organs still attached inside the body of a comercially cleaned chicken and I've seen more than a truck load of birds in my 15+ years of cooking professionally.

Pics of cleaned chicken

Here's another good pic for you. It's the concave bone of the pelvis that holds the oyster. When they cut the bird up for fried chicken they cut the bird in two, down the spine then make three cuts per side and seperate the dumb stick. One half way up the pelvis, the next one splits the upper body into a wing section and a breat section. This way you'll end up with 4 fairly equal sized pieces that should cook at about the same time.

I am not sure what the organ inside the chicken pelvis is, BUT it reminds me of, what in beef, is called the "sweet bread". They have one thing in common, they both taste like the liver of that animal, but with a lighter, more delicate flavor. I have also eaten pork and beef kidneys -- what is called the "gizzard" is closer in both texture and flavor to a kidney.