Which parts of a chicken are used to make chicken soup?
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).