None of the answers so far mention collagen -- specifically, using a stock made from roasted and cracked poultry bones.
You don't have to roast the bones, but you do need to crack them -- use a large, heavy knife or cleaver (not a chinese vegetable cleaver, you'll screw up the edge) to chop the bones up into about 2" to 3" pieces (5 to 7cm). Put them in cold water, and slowly bring it up to a simmer, and just leave it simmering for a few hours. You can add vegetables, too, but you'll want to pitch them, as they'll have been overcooked by the time the simmering's done.
Then strain everything, and use that as the base of your soup.
If your refrigerate stock that has enough collagen in it, it'll set up like jello, and hold its own shape. Starches can work, but they're not ideal for soup -- flour (unless cooked as a roux) leaves a raw flour taste and cloudy soup; tapioca leaves little granules in there; corn starch will break down if you cook it too long.
rfusca already gave very good suggestions for the literal question from the title. However, you can also address your problem the other way round.
First, cook the soup until your noodles are al dente (but will become just right while cooling at a normal speed). Take the big pot of soup off the heat.
Second, take a small pot, and fill it with just one portion of soup per eater. Put it on the heat, and cook until the noodles are done. (Alternatively, put the single portions into porcelain bowls and microwave until the noodles are ready - it spares you washing an additional pot, but I wouldn't nuke a good soup for no reason).
Third, eat your cooked soup portions and let the big pot of soup slowly cool on its own.
Fourth, freeze the soup from the big pot. Finis.
Note that from a food safety point of view, you are better off with flash cooling the soup. But what I outlined here is probably much simple and hassle-free. Plus, 2-3 liters of soup minus a portion or two should spend less than the magical 4 hours in the danger zone while cooling. If you are doing this with a very big pot of soup and feeding lots of people, you should probably prefer a rapid cooling.
Best Answer
similar to what julio said, the main reason is that typical wheat-based noodles release a lot of starch into the water, which changes the consistency of the soup. the starches can add a dirty colour to the water, but more than that, they can also thicken the soup undesirably (think of making a roux). Finally, if there are leftovers, the noodles can sometimes become completely water logged, making eating it the next day a bit of a soggy experience.
That said, with all these considerations there are times when you might add the noodles directly in-- non-starchy noodles (eg. like rice noodles) seem to do ok. also parboiling regular pasta before hand seems to help a lot with the starch and can still help you achieve the flavour integration you mentioned.