If you're qualified for something or allowed to take part in it, you have eligibility. Having a valid driver's license means you have the eligibility to drive a car.
The noun eligibility is good for describing the state of being qualified for or worthy of something. You might have the eligibility to take an upper level computer class, but not have the eligibility to take advanced Arabic, for example. Eligibility comes from the adjective eligible, which comes from the Middle French, meaning "fit to be chosen." The Latin root word is eligere, "choose."