You made three mistakes:
- Never tell an interviewer that they have asked you the wrong question.
- Never tell an interviewer that they have asked you the wrong question and list 3, 4, or 5 variants of the question that would not have been so wrong.
- Never tell an interviewer they have asked you the wrong question, list too many "better" questions they should have asked, and then just stop without answering anything.
That last one might have cost you the job. I can accept a certain amount of lecturing about how to interview properly and put it down to nerves and the usual complaints about "kids these days" and how the millenials are ruining the world etc (that's sarcasm btw) but if you don't end up answering something then you've blown it.
How should you have handled it? Probably something like this. I'll use italics for what you're thinking, and regular type for what you say out loud.
What was my biggest life mistake and how did I go about solving it? I'm not going to answer that question because I, I think it was a pretty bad question because, well first of all, I think it is too personal and secondly, if they are looking for a truthful answer, then I would need at least one minute to think about my previous life events which by the way have nothing to do with Web development and design. Let me think for a moment. I've faced some challenges and overcome them, of course, but mistakes that I've made... that time I slept through an exam, no, makes me look unprofessional, turning down the scholarship from the more famous university, no, I don't think that was a mistake I stand by it, oh got it! There was an incident last year where I felt underprepared going into the midterm. I talked to some other students and they said that nobody really felt confident in the material, so probably the exam would be belled, or maybe we would only be tested on part of the material. That sounded like a great theory to me compared to putting in the work to fully understand the material. But no, we were tested on all the material and the marks weren't adjusted. Now I have a lower mark in Applied Whatever than I would like. I should have gone to the professor or the TA and asked about the material I felt unsure of. I know I will in the future when I feel unprepared.
What makes this a good answer? First, you buy some time by repeating the question and explicitly stating you need a minute to think. There is nothing wrong with taking a minute to think - when candidates rattle off prepared answers I often detect insincerity. Second, you clarify the scope of your answer - you're not going to tell them about overcoming challenges, or your biggest weakness, you are actually going to find a mistake and tell about making that mistake and trying to solve it afterwards. Third, you do just that. You choose a mistake that isn't insanely bad (one time, I stabbed a fellow student, which in hindsight was totally a mistake) but that isn't trivial (I forgot to get ketchup on my hamburger) either. You tell a story of the appropriate length, which contains a minimum of excuses, describe the consequences of the mistake and what happened after that.
There are many bad answers to this question. Claiming to never ever make any mistakes is a sign of a delusion. Telling a LONG story about how totally unfair it was that you got a low mark when the prof was clearly to blame and your mistake was not trying to get the prof fired right away is also a red flag. Telling a very personal story may also be wrong. Some personal stories may be ok if they might relate to work - say you overpromised something and from that learned never to overpromise, or you missed a personal deadline like the last day to shop before Mother's Day and you learned from that to plan things in advance.
It takes some skill to answer a slightly different question than you were asked, but if that's your plan, do that. Don't give a big "you suck" speech with 3 or 4 options that would be better than what the interviewer asked you, and then trail off after offering to answer one of them. Just reframe the question and answer the one you'd be willing to answer. If (and I would be astonished if this happened) the interviewer replied, "no, I mean a life mistake like choosing the wrong school, cheating on your lover, getting in a car accident, drinking too much and getting arrested, that kind of thing" you could certainly give them the speech about being personal and not related to web development. If. But a better first reaction is simply to answer the question you'd be willing to answer, and answer it well.