Be sure to scroll down... there may be more than one question on this page!
You are going to have to hunt for the right material. Some of the better blind companies actually sell repair kits for their products. If you are unsure of the manufacturer, you will have to let your fingers do the walking (in the phone book) and call around to curtain and blind stores to see who stocks replacement string. You might even try a home store.
Don't make the mistake of trying to use hardware store string... it will not last. The strings used for many blinds are very strong for their small size, and are specially woven so they don't unravel easily. Also, the texture of the strings is smooth and more resistant to abrasion and wear.
The only exception would be the old style large wooden Venetian-type blinds. For these you may be able to get a suitable grade of string in a hardware or home store. Bring a sample of the old string for a comparison… just to be sure.
Since each set of blinds is constructed somewhat differently, I can't really coach you from here. You will have to summon up your handyman skills and analyze how the blind works. All of the blinds with metal top tracks disassemble by removing a cap on the end… usually the end with the string. In some cases, though, you may be able to do restringing without completely taking them apart.