Correct spelling, explanation: the word is created by the process of adding an adjective suffix -ous to the noun prestige. In the process of word formation, the silent e at the end of prestige becomes i in prestigious as it suits the pronunciation. The other theory is that it comes from Latin praestigiosis, which would explain the origin of the spelling even more directly.

Definition of prestigious:
Adjective
1. Being of a high standard and good reputation; being very highly regarded
It was a prestigious hotel, so only the wealthiest could afford it.
Our company won a prestigious award for our recent project.

Incorrect spelling, explanation: while it may seem like a word created analogically to gorgeous, it is not. Most likely it is caused by the possible Latin original word praestigiosis, which dictated the i in the place of e. Either way, despite convincing looks, prestigeous can not be considered correct.

Incorrect spelling

Incorrect spelling

Incorrect spelling