This semester I started working with a professor and a PhD student on one of their major projects, which is an application of their research. Though I was not involved with the research that went into developing the "idea" for the project, I have spent a significant amount of my time helping them to actually build the project, and a lot more work is planned.
Would it be typical for someone in my position to be included as a co-author on the paper that motivated the project I'm working on? Again, I wouldn't say that I have done any research myself, but I have done significant work in getting this project built, and the project is the main product / showcase of the research they performed.
Assuming it's reasonable for me to be included on the paper, how should I go about bringing that up?
It is really up to your advisor to include you as a co-author or not. If he feels that your work has significantly contributed to the paper, then s/he (without you telling him/her) should include you. Co-authors can have different responsibilities/roles when publishing a paper. One can have the IDEA, another will find the APPLICATION, the third will WRITE, the fourth will bring the FUNDING etc. Even if somebody was not a part of the research (for instance, friend's PhD co-advisor, did not really contribute much to his research but was a co-author in all of my friend's paper (3rd author to be exact)).
Also, if I was your advisor, I will ask you to write a/few sections (in addition to your work that you have done) to include you as a co-author. Giving that you did not do "research", maybe your writing needs work (it will a pain for me to go over your work/teach you). Still, this would be my way of encouraging you and introducing you to the publishing field.
After all, students need to learn, advisors need to advise/teach. Publications are the outcome of interaction, quality work and research.
Even if your advisor declines, ask him/her "respectfully" to be part of the next study. Tell him/her that you are willing to do some research because you would like to be a co-author. The more you engage, the better your chances you will have of being a co-author.