The main problem lies within the irrationality of the fear of flying. While there may be many reasons to be anxious about flying, none of them are "statistically reasonable", with the exception of unwillingness to travel by air because of motion sickness, which I do not think classifies as fear of flying per se.
Since the fear of flying is irrational, it is highly unlikely it can be overcome with reasoning. The subject has made a profound internal analysis that deems flying terrifying. Facts about the safety of air travel will not help, and neither do reassurances of the multiple layers of safety within the industry. The prognosis is not good, so to say.
Have you inquired your cousin what is the reason for her phobia? That may give you some clue as to how to approach the subject, keeping in mind what I wrote above. Above all, I certainly would ditch the idea of persuading her to fly with you as any sort of a goal.
My suggestion is to offer her a chance to familiarize herself with aviation at her own pace, with no binding schedule and, most importantly, a chance to bail out at any point (pun intended). This introduction should be done in steps, that can, and preferably should be distributed to separate occasions:
- get her to visit the airfield, explain how GA stuff works there. Do not talk about flying yet, unless she specifically asks about it. Explain ATC (even if your field didn't have it), who operates the field, how many people buzz around. You know, general stuff, sorta like in a Richard Scarry book.
- introduce her to the planes on the tarmac. Their features, why they are different, and why they are same etc. Dig in a bit deeper into the subject than last time. Show your expertise, but don't go in too deep.
- get her in the cockpit, show and tell what's what, and why. Gauges, yoke, pedals, flaps etc. Don't rub in her face, but every now and then, when appropriate, drop a few lines on the safety aspect of doohickeys in the plane, such as "we use flaps during approach and landing because that makes it safer to fly slowly".
- show how a preflight is done. Like a proper preflight with planning.
- if/when she's comfortable take her taxiing around the field. If ATC is present and traffic situation allows, they'll surely let you even taxi to and along the runway for familiarization.
- should all go well, she may be ready to take a flight in a while. Make a small sightseeing flight, pick a nice familiar place to go and see, something she can relate to safely. Do not fly closed circuit with a person with fear of flying.
Main point is to explain to her you have this plan with the general steps above, and that at any point she can stop, and there will be no persuasion, picking or anything negative. No strings attached.
If your local field has a cafe or a good sighting area, that's a very good starting point: go planespotting first (and secretly hope no-one effes up while you are watching). If there are night operations, do a night picnic, airfields are mesmerizing places by night. You should aim at making flying more interesting and exciting than it is frightening.
And remember: if her phobia is anything but mild, it'll most likely be way out of your league to fix. Fear of flying is usually quite impermeable.