Help! My indoor garden is getting overrun with the pests! What's a good home-made solution?

Insecticidal soap. Spray every 5 to 7 days until the pests are gone since what you are trying to do is interrupt their life cycle, generally the eggs hatch in about 5 days or surviving aphids produce new, live aphids about then.
To make your own insecticidal soap put 1 teaspoon of a real soap (one made from animal fats, not the detergents) in 1 quart of water. You need about a 1 percent solution of soap to water.

Mix one ounce of liquid seaweed in a half gallon of water and spray.

Thanks guys for the answers! I got the insecticidal soap and that seems to be working for now. I will try the seaweed solution next time.