I had one that did something similar. The screen lit up black, it made a weird short noise, then turned off.
In my case, completely disassembling and reassembling the 3ds fixed it permanently. I'm 90% sure it was a loose ribbon cable or connection somewhere. You should try taking it apart (Make sure there isn't any apparent water damage), cleaning it, and putting it back together. Pay extra attention to the ribbon cable for the screen. I think that's the most likely culprit for my issue.
On a separate note, if you plug it in, does it charge okay? If it won't charge it could be as simple as a blown fuse which is a 10 minute fix for a ds lite (don't know about 3ds)