How to analyse and prove to business stakeholders that a website redesign (b2b hardware) is better than a previous one?

Use Delighted: https://delighted.com/

Are you performing a large-scale customer study? Or is this just a smaller focus group gig? The primary rationale for hiring a user research recruiter would be to obtain a valid, representative sample for statistical inference. If the number of participants is small -- too small to be statistically significant -- then you cross this concern off your list. Presumably, you know who your ideal users will be. After all, they must be your future customers, right? So if you can't track them down now, then future marketing will be pretty hopless. You can probably contact them on your own without an intermediary. Another reason for hiring a recruiter might be to inject a buffer layer for impartial selection. Otherwise, you and your colleagues may inadvertently bias the selection process toward people who know you or are already too favorable. My experience in this area is not vast -- although it sometimes comes into play for larger corporate naming initiatives. So I may be missing other arguments in favor of hiring research recruiters. I'd be glad to hear those from other people so that I can learn something too.

I would start with google analytics. If you configure it correctly you can get quite far down the line for free. Once you start seeing revenue and need more detail you can move over to one of the more premium or paid for systems.

I would disagree with the premise that "big data" isn't being converted into valuable information. Modern advertising is being driven by exactly this ability. The problem lies not with the information, or lack thereof, but with the decision makers. Lack of vision at both the strategic and tactical levels of decision, fear of change, among other things are at the root of the problem.

The best way is to lean how to ask great questions and to physically spend time with your users watching them work. I don't have all the killer questions, but my favourite to help understand product roadmap opportunities is "What do you do 3 minutes before, and after you use our product?". That helps me understand where the opportunity lies to build a solid solution end-to-end. Also, finding a set of users that feel comfortable having you hangout and work with them, taking notes and asking questions is invaluable. Just to understand their environment, the way they organize their work, and how they might communicate around the problems your solution is/could help with is incredible. Every time I'm at an event and someone ask me what I do, I always ask them if I can "show them" and if they say yes, I have them find / download / signup and explore our app. Then I ask them questions about how they might see someone using it, who they think might be the best customer, etc. It's not about not knowing, it's about continuously learning and testing different perspectives. As for tools I use: www.usertesting.com / especially new mobile tests!