There are three main factors you want to consider when purchasing a pillow for a child: its size, its ability to be cleaned, and the reputation of the company from whom you are buying. The first is a necessity; the second and third highly recommended.
Regarding size: Please follow the pediatrician recommendation for a SMALL pillow, one designed to fit their little necks and bodies. Large, adult size pillows are just not healthy for little people. If you want to learn more about correctly sizing pillows for children, read this article.
Regarding laundering: Ironically, not all children's pillows out there can be washed and dried so be sure to double and triple check the details. Children get sick in the night and sweat and drool far more than adults and therefore will need a pillow that can be regularly laundered. Pillowcases certainly help mitigate the normal wear of a pillow but this won't stop the extreme moments that every child has at some point during their early years. And be careful about the pillows that claim to be resistant to these moments (this includes dust mite covers) as they most often are made from a plastic cover which is extremely hot for a child to sleep on, not to mention not that comfortable. And for those who are worried about the 'dust mite allergy', any pillow that can be regularly machine washed and dried will take care of that.
Lastly, be sure to choose a pillow from a reputable company. There has been a flurry of popup companies in the past few years cranking out children's pillows because they believe it's a quick way to make a buck without any real thought or care for longevity and quality. We have seen many of these copycat companies' products and have been appalled at the low workmanship and lack of real understanding about what makes a good quality, enduring pillow.
As a personal appeal: please consider buying a truly American Made pillow. Again, many of these hacker companies are claiming to be "made in the USA" when they are only assembled in the USA. This is a big difference from actually making the product here in the US vs slapping a last minute tag on it here in the USA and claiming it's made in the USA. In the long run a few extra dollars for a truly American Made product will be money well spent.