In the middle of a project that's gone way beyond my estimated timeline, the client said they want to pay me more. I don't feel comfortable taking more money since I underestimated the timeline. What do I do?
Sometimes people with more money than time value high quality work and want to pay what it's worth.
If a project goes beyond its initial scope, I wouldn't continue without discussing that directly with the client. Without you speaking up, once they noticed the additional work, they probably felt bad about it and maybe even a little frustrated that you didn't speak up. This illustrates the importance of open communication, even when things are going relatively well.
It's ultimately up to you whether you feel comfortable taking the extra pay. Maybe it depends on how badly you underestimated and if you can afford to take the hit. Or you could use it as a learning experience, show good faith to the client, and say, "You can get me on the next one."
Whatever you do, discuss things like this with your client early and often. That's the only way to a happy resolution.