What is a suitable job title for someone responsible for data and software in a research project? My PI has just asked me what she should call me when introducing me to visiting researchers. I couldn't give her a definite answer.

I work in a research project where I supervise its research data in terms of versioning, formats, conversion, storage, etc. In this function I've, e.g., written a data management plan. I also (and this is the gros of my work) develop software which is used within the project but is also generic enough to target other researchers. And finally, I work on my PhD thesis, albeit not in a structured PhD programme (in my neck of the woods you can write a thesis as long as you have one or more supervisors with the right to supervise PhD theses.

As of now, I've usually been introduced as the "project's programmer", which my PI finds is somewhat dismissive of my actual work and responsibilities.

What would be a good title then, one that could also be printed on business cards?
This sounds essentially like my position: research programmer

It encompasses a lot, but perhaps 80% of it is helping researchers who know what they want to accomplish but have no idea how to get there.  It can include things like web scraping or cleaning up very ugly code, or translating a project to a different codebase, or writing simulations, or running models.  Version control often comes into play, along with data storage - I'm often teaching researchers about GitHub or S3, for example.  It also tends to involve data vis and data presentation (like on the web).

It also has an element of data science, where a researcher gets access to some data that has what they want but is too big for them to know how to work with or manage.  But I wouldn't call yourself a data scientist; it's a whole separate field with its own methods that often diverge from traditional research.

It sounds like "research programmer" encompasses everything you're doing, but sound a bit more grand and more accurate than what you're using now.  It's also one that is in official use elsewhere (in my case, the policy and think tank world), so that's a bonus.