I boiled a chicken legs for over 2 hours in low temperature. The soup is yellow and with very soft chicken. It tastes very good after adding some salt. Which part of chicken is the most appropriate for doing a soup? I am looking for the lowest cost option.
The lowest-cost option is definitely bones, assuming you don't actually need or want pieces of meat in the soup.
When I say bones I'm referring to the parts that are normally thrown out as waste products, especially the necks, feet, and carcasses. If you can't find them at a supermarket, talk to a butcher. They are sold for next to nothing - usually less than $1/kg, sometimes as low as $0.50/kg. That is even less than the per-unit cost of a whole chicken. Some butchers (no promises!) will even give them away for free, or at least haggle a little bit, especially if their business is small.
You can roast these and simmer (not boil) them over a period of several hours to produce a very rich stock. Usually you would do this with a mirepoix and/or bouquet garni for flavour. When it's done, you strain it, and then add fresh or frozen vegetables, noodles, etc., to stretch it into a full meal or at least a hearty soup.
If I really wanted meat in the soup, I'd opt for thighs, as they are inexpensive and (once butchered) you can throw the bones in with the rest.
Other than pure bones, the "budget" option for chicken is almost always the whole chicken, if you can use it all. That means doing your own boning and butchering, then using the bones and trimmings for stock/soup and finding ways to use the offal (combs, hearts, livers, etc.) The offal really isn't any good in soup, so if you just want chicken soup, then stick to just bones and one inexpensive cut (over here it's thighs, but this may vary by location).
The tradition in Scotland, my homeland, is to cook the whole chicken in water. That's feet and head and all (if the butcher leaves them on) along with some leek and onion. When the chicken is finished cooking, don't cook it too long, you remove it from the water and cut off the meat. The meat can then go to make your main course meal while the cooking water becomes broth for soup. You can throw some of the little fiddly bits of back and neck meat back into the soup to add flavour.
Once your removed all the edible part, put the bones, head, feet and other bits you didn't consume into another pot and boil again. You'll get a second batch of soup broth from this. You might even get a third batch though this will not be as strong as the first ones and will take more boiling.
This way you have efficiently used every part of the chicken.
For a soup, you want slow cooking meat with bones. Legs are very good. The hips and back are also good. Breast (white meat) will become dry and stringy. Wings make good stock, but don't have enough meat to be good for eating. The tail piece has too much fat. The innards are bad for a soup, they are very prone to overcooking.
I don't think there is much cost difference in chicken parts. Steaks from the white meat cost more, but there aren't bones in them, so they are a better value. You can buy legs only, or buy a whole chicken and use its legs and hips in a soup and the breast fried. If you get the innards with it, fry them too. It is your choice where to use the rump meat, but for cooking times of two hours, I'd rather fry it too. But usually, I cook soup shorter.