An idioglossia is an idiosyncratic language invented and spoken by only one person or very few people. Most often, idioglossia refers to the "private languages" of young children, especially twins, the latter being more specifically known as cryptophasia, and commonly referred to as twin talk or twin speech.
Lots of material here. I already keep track of what my boys say, how they phrase things. One of my favorites was Wystan describing something good as being gluten-free.