Cheap hair dryers aren't intrinsincally bad products, and you can certainly find some diamonds in the rough if you know what to look for. The main things to consider when buying a hair dryer are; the maximum temperature, whether it has ionic and ceramic heating elements, and it's weight. These details can be harder to find on budget hair dryers, but searching them out can make a huge difference.
Both the temperature and the heating elements have the potential to damage or smoothen your hair depending on which side of the spectrum they fall on to. Temperatures above 60 degrees C can cause damage, with irreversable damage starting at 140 degrees C. The best hair dryers tend to have in-depth specifications noting the temperatures for each heat setting, but cheap hair dryers don't. Without this information you need to look at wattage as an indicator of how hot the heating element will actually get. The higher the wattage, the hotter that top setting is going to get to, and the more likely it will be to damage your hair. For cheap hair dryers, look for a wattage range starting below 2000W where possible, and promise that you'll never use the super hot setting. PROMISE ME!
You'll also want to make sure the hair dryer has some sort of ionic or ceramic element. These create an extra layer of protection between the hearing element and your hair which can help minimise damage. Ionic charges in particular can tame static and leave you with a smoother look.
Budget hair dryers tend to be on the heavier side which can make them quite uncomfortable to use for extended periods of time. Unless you want an arm workout with your blow out, look for lighter devices.