If you have access to a farmer's market or fruit/vegetable stand your sides can be really cheap. Winter squash, brussels sprouts, apples and chestnuts are "in season" in Ohio right now and potatoes are available, too. There's a small farmer that sells veggies out of his garage. I got a peck of apples for $4 (apple pies).
I grew my own squash this year, but he was selling them for $0.75 a piece so you can serve squash instead of yams. Also, I've made "pumpkin" pies with cooked butternut or buttercup squash instead and no one knew the difference. Every year I make stuffed squash (I'm vegetarian, so someone else handles the turkey).
Using stale bread for stuffing, as someone else suggested, is a really good idea. Check the food network website (foodtv.com) for recipes, or the veggies unite (vegweb.com) website.
I also make this dish every year, and people really seem to like it. It's not my own concoction, but I can't remember where I got the recipe. Slice brussles sprouts in half. Heat some vegetable oil in a pan over medium heat. Place sprouts, cut side down, in pan. Cover, reduce heat to low, and cook until sprouts are softened. Remove pan from heat, add a few pats of butter (it is a holiday, after all) and serve warm. It's really important to make sure the oil is kind of hot before you put the sprouts in it, otherwise they absorb a lot of the oil while it's heating and end up being greasy. You can put roasted chestnuts into this dish (and/or the stuffing). It seems to make it more festive.
Making your own cranberry relish out of fresh cranberries is a good idea, too (someone else suggested it). I have friends that put the leftover relish on turkey sandwiches (made from leftover turkey, of course). I use it in muffins for breakfast or stir it into oatmeal while it cooks.
Finally, remember that holidays are about getting together with friends and family. Thanksgiving is supposed to be about being thankful for what we have. If you can't afford to make or buy a specific dish, replace it with something cheaper or just skip it. I'm sure you'll have enough food to feed everyone and if a particular dish is that important to someone ask them to bring the dish (or the ingredients). These people are your family, they should understand that your budget has limitations and you will do the best that you can, so don't sweat it. Besides, what's the worst that can happen? They think you blew it, so from then on you have to be the invited guest who doesn't have to prepare the meal, clean up the mess, entertain out of town guests, etc. How awful
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I hope you have an enjoyable Thanksgiving!
Leah