Many, many students who are at lower-ranked universities aspire to be at higher-ranked places instead. (In fact, what is more surprising to me is that not all such students feel this way.) The hard truth is that top-ranked universities by definition are very selective of their applicants, and in your case the "main round" of the selection process has already occurred. It is not impossible to transfer to a much better place, but it is difficult, to the extent that it may or may not be a reasonable goal. Growing where you are planted sounds like a smarter strategy to me, and in fact it is not fundamentally incompatible with the ambition to move elsewhere.
The one way I can think of to effectively demonstrate your potential is to realize it, i.e., to prove at least one theorem of a caliber beyond what most students in your program do as part of their PhD thesis work. If you can do that, then you clearly deserve to be at a better place, and you should not have much trouble finding some place to take you on. I don't think anything else is worth banking on: as others have said, grades in PhD-level mathematics courses in the US are not necessarily very meaningful. This varies in its extent, but every program I know has enough courses where all students who do not do anything egregiously, ridiculously wrong get the highest possible grade that a 4.0 GPA at the graduate level is just not the same thing as it is at the undergraduate level. I would expect that a substantial proportion of the students in your program would have 4.0 GPAs.
Unfortunately I think the strategy of "gunning for an early research success" will either be a good one or (and in terms of the percentages, this is more likely) a very bad one. Most early career math PhD students are not anywhere near equipped to do meaningful research, to an extent that other academic fields would have a hard time believing. In my program (about the 50th best in the US) early career students neglecting their coursework in favor of their "research" is a big red flag for me: these students often fail to meet programmatic requirements and get in real trouble.
To be very honest: I think the big risk here is that you are not as exceptional as you think, that your lack of effort in your coursework will itself become a problem down the road, and that your superiority to those around you may only be a matter of opinion. Of course I'm not saying that this is the case -- I couldn't know -- but it would be very smart to be concerned about this on your own behalf. You need some reality checks. Here are some checkpoints:
Step 1: Can you pass all your generals / quals / required coursework rapidly and unusually well?
In most non-top math PhD programs I know of, there is an initial phase of 1-3 years where students spend time studying coursework and then passing exams. A small number of students get through this portion much more quickly: within a single year, a single semester or even immediately upon arrival. It is debatable whether such students are truly more talented than the others or simply arrived very well prepared and have their act together, but in any event such students are certainly doing exceptionally well in the program.
Step 2: Can you get faculty on board with the plan of your having exceptional promise and ambition and get a very meaty project to work on?
Go to one of the faculty members you're working with and level with them: say that you feel understimulated by the program, are worried that your peers lack your intensity and ambition, and that you are really looking for a project to sink your teeth into. Then step back and listen very carefully to what she has to say. You will probably get some amount of corroboration/denial of the picture you have painted, though it may be subtle. (I would not say: "No, I completely disagree: you're just average, maybe not even that" to a student, even if I felt that way!) More importantly you should get something to work on. Do so. If it is not what you wanted, or if you have your own ideas about what you want to work on, come back and say so, but I think it's good to spend some time on projects that other people give you at this stage.
Step 3: Can you get some results? Anything at first, then something nice, then maybe a real breakthrough?
If you can make it successfully through Step 3, then as above you're ready to go somewhere better. Let me not even try to hide the following key point: the merit of this checklist is that it's what you're supposed to be doing anyway. If by the time you successfully complete Step 3 you're ready to graduate, should you curse my trickery? No, you should proudly write up your thesis and expect to get a postdoc at a much better place than you did your PhD, which is hard to do and puts you on a great trajectory for your later career.
Good luck.
