Waste not, want not is a saying that someone's grandmother always said. However, it definitely has merit. Again, I hate throwing anything away. There's something inside of me that screams, "BUT YOU COULD DO SOMETHING WITH THAT." Sometimes I force the little voice to shut it. Other times, I listen, and then I have something available when I need it.
I'll be honest, I love my cardigans. I have one in black, oatmeal, purple, navy, grey, ... and I'm in the market for a yellow, red, or green one. They're not cheap items, and rarely do you find them at GoodWill, but sometimes you do. It saddens me when they get a small hole in them, but with a little sewing machine magic and a cut up old t-shirt, I have a cardigan that looks new again.
Prep: Go to your local fabric store and match your cardigan to the thread they have there. More often than not, they don't mind if you take a tail end of a thread out of the casing and make sure it matches. At home cut a square of t-shirt material that is big enough to cover the hole on all sides from the inside, with extra on all sides. Don't forget to wind your bobbin with the same color!
Let's do it: Place the t-shirt fabric on the inside of the sweater, set your satin stitch to where it will cover the hole, but not much more than that. Take great care to only sew the spot that needs mending. After, cut the t-shirt fabric down making sure not to cut your newly mended sweater. Whew, a few dollars and a little effort just saved your sweater. It's worth it.
NOTE: Old t-shirts make excellent reusable tissues. Place 2 same side squares right side to right side and stitch with 1/4 inch seam almost all the way around. Leave enough open to turn the square "right side out" turn in the unfinished edge and seam. Easy, and a great way to not throw something into a landfill.