I'm not sure whether this piece of advice is generally useful or not, and it's almost certainly variable depending on field; but I do wish I had followed this when I was in graduate school.
Fairly early on, choose a thesis topic or problem you're interested in, and pursue it. Don't get side-tracked into something just because it's what the department or your advisor is working on. In fact, I'd go so far to say you should pick your thesis topic before you apply. Use that to help you decide where to apply. If you don't have a thesis topic you're committed to, you're not ready for grad school; and should stay out.
If you can swing it, get informal approval for your topic and an agreement from a faculty member to serve as your advisor before you apply. This is easier if you're attending grad school the same place where you're an undergraduate, but it can be done as long as you have some notable research chops as an undergraduate.
I heard the opposite too: don't try to push your own research too hard, especially in the beginning, but try to find something the others in the lab group are working on, learn stuff from them, and after a while you'll end up with your own ideas anyway.
Pros and cons for both of the approaches exist, but "picking a thesis early on" might be hard as you don't necessarily know what the good problems are in your field. But that might depend on your field / advisor too.
There was some support for the idea of starting an advice repository for grad students much in the same tradition as the Boring Advice Repository and the Solved Problems Repository started earlier by Qiaochu_Yuan. So here goes.
Please share any advice, boring or otherwise, for succeeding at grad school. I realize that succeeding might mean different things to different people, but I believe most people largely agree with what it means in this context. Feel free to elaborate on what you believe it should mean, if you have views on the subject.
I am a theoretical physics grad student, so I'm personally more interested in advice for mathy disciplines (i.e. physics, math, CS), and I also suspect that there are many grad students from these disciplines on LessWrong; but advice for any discipline is welcome as well.
Advice is welcome from anyone, but please do mention your background for providing the advice so that people can weight the advice accordingly. For example, I would be more be open to listening to advice from someone who has completed a very successful PhD, than from someone who has simply interacted with a lot of grad students but has never been to grad school.
Also, feel free to link to advice from other sources, and maybe quote the most useful parts in what you read. Remember, this is meant to be a repository, so that people can come and find the advice, so don't worry if it seems to be something most people would've already read or known.
Thanks!