Children are very resilient. Their bodies heal faster, and more efficiently than that of an adult. I can't tell you if it will clear up or not. However, kids are less likely to scar than an adult.
There are some things you can do to increase the chances that the bed bug bite will not fade into a permanent scar.
1. Clean wound with warm water and use antibacterial creams. Neosporin is a great choice, They have new products that are even made to prevent scars.
2. Use Calamine lotion to prevent the itching.
3. Buy a scar removal cream. There are real products, usually made with Vitamin E and other active ingredients, that reduce the appearance of scars. I use Mederma, but there are lots of great brands out there.