Vega is not a greek letter. However the mathematicians Fischer Black, Myron Scholes and Robert Merton, included it in their theory of option pricing that led to the emergence of the renowned
Black-Scholes model and it has stuck ever since as an 'imaginary letter'. It is unclear why vega was chosen, as all the other parameters it is related to in
Black-Scholes are denoted by standard greek letters. It is most likely that it 'sounded' right and its use has probably persisted due to the practice of options traders using it due to it starting with a 'v' just like the
volatility which it helps to describe in the model.
To make matters worse depending on the various literature you read 'vega' is on occasion replaced with any of the following conventional greek letters; kappa, tau or lambda. (It appears that each new contributor to the theory of the option pricing had their own convention). In my notes taken from old university classes vega is written to resemble the small greek letter nu, ν, but with a further squiggle on the right-hand side of the letter, like you couldn't be bothered to lift your pen.
In Black-Scholes vega is used to measure the change in an option price in relation to a change in the perceived volatility of the market (i.e. risk). In broad terms a small value of vega implies that your model has a better estimate for the volatility of the market then for a large value of vega. This is because most models in the short term assume market volatility to be constant.
Asides from the origin in Babylonian astronomy as described by Dreamvirus above, Vega was also the name of a brand of Chevrolet car sold in the 70s, and is also used to describe a tobacco field in Cuba.