Idioglossia

Idioglossia \id-ee-uh-GLOS-ee-uh\ , noun;
1. A private form of speech invented by one child or by children who are in close contact, as twins
2. A pathological condition in which a person's speech is so severely distorted that it is unintelligible

The etymology of this word is really not that interesting: idio- is used to indicate the individual or something attributed to the individual and -glossia is spoken language. The really interesting thing is it's implications in the tabula rasa/innatism debate, which is a language-specific aspect of nature v. nurture.  
Tabula rasa means "blank slate" and it was introduced by John Locke as a theory for how we acquire language. He argued that we are born 'blank' and language imprints on us from what we hear as a small child. Innatism is the opposite argument, stating that we are born with a structure for grammar (sort of like a rough outline) and we insert the language(s) we hear into that existing framework.
There are compelling arguments for both sides, but in my opinion there is just a little too much science backing innatism to really buy into tabula rasa. Take our word today, idioglossia, and what it refers too: one could argue that these are just altered versions of language that one has already learned, and they would have a point. However, this is not the only possible type of idioglossia. Consider born-deaf individuals who were not exposed to a formal sign language early in life. These people are deprived of all spoken language in their formative years, yet they are demonstrated to be able to communicate complex thoughts with others through forms of sign language they have developed over time. The famous case that linguists study is Nicaraguan Sign Language (NSL), which was spontaneously created in the late 1970's and early 1980's in a school for the deaf. Prior to this time, deaf Nicaraguans were largely isolated and depended on 'homegrown' signs to communicate their needs. In 1977 a program was created for deaf students which quickly grew, serving over 400 students by 1983. The teachers attempted to connect with the students by using a finger-spelling system to teach them Spanish and lip reading, but it failed miserably. The students, however, started to develop their own way of communicating with each other which quickly took on the appearance of pidgin languages. In 1986, MIT linguists saw that younger students were starting to add grammar and conjugation to the pidgin of the older students, which is recognized as the second stage in the development of a new language.
Today NSL is considered a full-fledged signed language, though it is "unwritable" and "unspeakable", and it is still heavily studied by linguists because of it's unique status as a newly created language.
I would argue that NSL was formed when idioglossias between students began connecting to the idioglossias between other students until it became a large web that became increasingly interconnected and complex until it reached the status of pidgin and continued on until it became an independent, unique language.

*Today's word and the first definition were both taken from Dictionary.com's 'Word of the Day' for Wednsday, November 17