I hate to say it, but choosing a BI tool is like choosing any other software - you have to go through the same process of getting stakeholders together, writing logical requirements, then writing functional requirements (the criteria you were talking about), then moving on to demo software and shop for a system.
I'm a data scientist, and when I did this in the early 2000s at a small, non-profit research institute, it wasn't a very complex exercise, but it did involve me talking to different researchers and nailing down logical and functional requirements. At the time, I put together a system using SAS, R, SQL, but today, I assure you the same requirements would be met in a very different way!
Feel free to e-mail me or connect with me on LinkedIn if you want any specific advice. Have fun shopping!
Some important factors to look for are scalability, security, real time data, and good customer support. For more guidance, I'd recommend checking out this article 6 Features to Look For in a BI Platform.