Bill them for every single second of work. Perhaps it's because I come from a family of lawyers, but when that happens the customer simply receives a bill saying
Research: 40 hours.
Research: 40 hours.
Lawyers track every second but they don't bill for every second. In fact, a lawyer might mark-down hours for the very same reason as the OP -- something has taken longer than it should.
I am not blessed with infinite time. I am not doing this for you because I'm a nice guy. I'm doing the work because you are paying me. That is the entire deal.
If you are on the fence watch this: https://creativemornings.com/talks/mike-monteiro--2/1
You are doing your client a huge disservice by hiding hours you actually worked, even if you don't charge for them.
In the end you have the best intentions for your client but can end up making it worse for yourself going forward. Quote down to the minute you've been working on a project, if the client is un-happy then that is something you can work with.
If the problem you're solving is more of a general dev task, which 95% of tasks are then it's like a lawyer that is merely processing immigration paperwork. You're paying for it to be done right. Billing for research when you're writing CRUD apps is a good example of this.
If you're a Robert Shapiro or some hotshot taking on special cases, and winning, then you can probably bill for research. Billing for research when you're John Carmack qualifies here.
So I think the lawyer comparison is 100%. You should bill for every second. Just make sure you're not on par with the lawyer equivalent of needing to do research when building a CRUD app and you won't have any problem.
I think a better approach would be to charge per day.
At a certain point, you need to a baseline competence or you shouldn't even be taking the job. If people balk at you for not being ready to even take on the basic job title, that shouldn't be a surprise.
I'm not saying if you come across a corner case that's special you shouldn't charge for it. It's like if you had an exceptional immigration case that required research, not just filing form 1072B.
It's something to keep in mind because you can lose your reputation or clients being silly with the "research 40 hours" act. Sometimes losing a little money today is a win longterm. A lawyer billing research for basic stuff or nonsense that no one else would (learning on the job, basically), would and should lose his clientele.