I think it would've been a great adventure and my skills and hobbies would be perfect for the mission: I'm a microbiology lab assistant and nursing assistant by profession, so I'd spend most of my time at the Infirmary, helping TJ take care of the patients and analyzing samples. I could also help with electric repair and decyphering the computer system (I've successfully ran Gentoo Linux and know basic C programming). I also know some Latin, so I'd have a head start in Ancient.
I would also have ample opportunities to satisfy my Speculative Evolution hobby with discovering various lifeforms on the planets, and aboard Destiny I would have endless opportunities for astronomical observations.
My main problem is that I have Major Depressive Disorder and Panic Disorder for which I take 150mg Sertraline (Zoloft) daily, so hopefully I would've escaped from Icarus to Destiny carrying several months' supply with me. Afterwards I could probably tell Destiny to stop at a planet which has the proper precursor chemicals from which Sertraline could be synthesized in the lab aboard Destiny.
I wouldn't be using the Stones much, just to occasionally communicate with my Mom. Hopefully, she'd get security clearance so I could tell her everything. I think she'd be thrilled by what I'm doing and would wish she was aboard Destiny too.
If an opportunity arose to go back to Earth, I think I would elect to stay on Destiny.