Remember that it's really a conversation, and it goes both ways. They want to learn about you, but you need to learn about them as well. Are these people you really want to work with, and are you the kind of person they want to work with. This means you have to show your full self - honest and open, and expect the same from them.
Some tricks here include making sure to turn it into a conversation, and engage the interviewers - something like finishing up your answer with "and, have you faced that problem, and how did you solve it?" works well to make them think, and engage them. You can then have a discussion about the problem and really allow yourself to show how you think and engage with someone. It stops being an exam/quiz setting.
In a similar vein, you want to answer in a way that shows you've solved the problem, or something similar before, in a real setting. So, rather than just providing the book answer, you can say something like: "Well, that's a similar problem to one I had when I was at X, and we solved it by doing Y, and my responsibility there was Z.". This shows that you were genuinely part of the solution, and can work in a team, and have real practical experience that's worth hiring you for.
I find it helps as well, in both the above scenarios, to highlight the problems you had, and how you overcame them. Especially in a technical field, the majority of what you'll be doing is problem solving, so demonstrate that ability.
Use open ended questions to demonstrate what you're passionate about and know really well. Call it out and explain what in the exercise you really enjoy, then explain your most recent tips, techniques, and things you've been experimenting with. Passionate people come across really well, and it's much easier to talk about something you love, it removes some nerves and mental blocks. It also them demonstrates (again), your capacity to learn, grow, and improve.
All of the above help you bring your humanity, as well as you tech skills, and show you can be a great help and contributor to the team they're hiring you for, and get you away from having to just rote repeat book answers to standard questions.
Finally, some smaller items:
1. If you need a second to think, have a drink. I always try to make sure I've got at least a bottle of water, so take a sip, and it'll give you a minute to think, to breathe and to relax. 2. You can always just say, let me think about that, and ask for a whiteboard, notes, etc. So many candidates we interview don't, and it hurts them. In the past I've taken to pushing people to take notes and write things down to keep their thinking in order.