Some random thoughts for what I'd look for if I'd apply for a PhD position again. Some of these are questions you have to ask directly, some you can't ask directly without being rude, but you can check the lab's homepage to approximate answers.
1) What is the average time from start to graduation? This varies strongly by country, with some ridiculous years in the US. Outside the US I'd be wary if it's >4 years.
2) How often does the lab publish per year? If 10 people work in the lab and the lab publishes only one paper in a mid or low-tier journal per year I wouldn't join
3) What's the work/life balance? Does the PI expect you to work on weekends or other ridiculous times?
4) What's the funding situation like? Is there a chance they'll close the lab after 3 years? Is there a chance that your own PhD won't be funded, or that funding stops?
5) Is the PI/supervisor nice? Are the colleagues nice? What does your current supervisor/mentor say about the lab, if possible?
6) Can you take parenting time off? Does the supervisor/PI have kids of their own?
7) Is the lab's focus on applied bioinformatics or methods development? Many bioinformatics labs focus on one of those two, choose the one you like more.
8) How many other bioinformaticians work in the lab? I'd be VERY hesitant if it's just going to be you, you won't learn much and you'll have no chance to work on your own projects. Sometimes people only want someone cheap to analyse their data, all you'll get out of it is mid-level authorship, not enough to start a career
9) If you're into open source, does the lab publish open source? Same for open access.
10) Can you find a lab alumni and ask what they thought of it? (mileage may vary, sometimes you'll find someone who just didn't click personally, giving you a image that's worse than it actually was)
11) How many current/past lab members have side projects, run workshops, mentor etc.? Indicates a freedom to pursue own interests
Edit:
12) Is there an industry component, or some other non-academic part?
All good points but they should not all be weighted equally. I'd add that choosing a computational PhD is not different than choosing any other PhD. What matters in the end is how it's going to help you achieve your career goals. For me, important points should be the quality of the mentoring (related to point 8) and of the scientific environment and whether you get along with people in the environment you're going to work in. Papers are important for an academic career, less so if you don't want to stay in academia. It's important to do your research on the PhD program(s) and lab(s) you're interested in. If you can, try to figure out where the recent alumni have gone (some programs/labs have this on their web site).