I am graduating and applying to MD/PhD programs in June. How should I fill the gap year?
Fill the gap year with some research, but know that it'll only have impact on you once you get to an interview. Do lots of fun stuff - travel, etc!
I don't want to leave this professor in the lurch though, since I have my own project. Also, I want to know the results (and rabbit holes) of my project, so I would prefer to stay to see it out.
Stick around because you enjoy the lab environment and that's what you want to work on when you wake up in the morning.
Do NOT stick around for the professor's sake and do NOT stick around because you want to see the results. If the professor was desperate for results on your project, it would have been given to a capable postdoc. They know undergrads' time and, to be honest, skills are limited. They're giving you sometime to get your feet wet, learn how to do science, and if you get results, added bonus. She/he will be thrilled if you work a bit longer, but wasn't truly planning on it since you're graduating. Every undergrad/gradstudent/postdoc leaves their PI in a bit of a lurch because they're the current expert on their project and the project is never finished.
Which leads me to, science takes way longer than you anticipate. Sure, you think you can get results in a gap year, and maybe you can. Awesome. You also can run into a myriad of problems and be just as far in the project a year from now as you are a month from now (as any grad student can attest). If the latter is the case, will you still be happy with your decision to spend a year there? If so, stay.
There will always be loose ends and you'll never get all the answers. You'll need in reality a decade to answer the questions that have peaked your curiosity now and then you'll have uncovered a whole host of new questions during that decade. You only really get to explore and answer questions satisfyingly as a PI over a LONG time frame and even then you're limited by grants that get funded.
Remember the goals of what you're doing: you're learning to be a scientist. If you think you can gain useful skills or knowledge (engineering bent or whatnot) elsewhere, go elsewhere. You can literally go anywhere in the country (or world if you have the $ to deal with international flights during interviews) to do research this year. Being exposed to different lab environments, management styles, research focus/methods will only make you more informed as you train to be a scientist. If you feel like you have a lot to learn from your current PI, stay.
Taking a gap year gives you a lot of flexibility and have experiences that are difficult to obtain during the MSTP train once you get on. I"m a big advocate of taking advantage of it! I always remember the story a postdoct told me after he finished his PhD many yrs ago. He decided to stay around lab for the whole summer before he went back to med school because there was this one loose end, he knew he could complete it and get another publication out of it. So, he spent the summer slaving away in lab. At the end of the summer, he had nothing to show for it. In retrospect, he really wishes he took at least a month or six weeks off to recharge before heading into the med school fray because he really had no responsibilities. And a close to a decade latter, the problem he was working on was SO irrelevant to his current research and he wished he was less burned out starting back in med school.