I know this Is old but it MAY be possible that I can help,experienced with a 3DS from launch. [br]
First one to try is if your using a Micro SD card with a adapter,try and remove it from the adapter and clean it off and blow out the adapter port.[br]
Another one to try is removing the battery and holding the power button for 30 seconds and re-inserting it afterward.[br]
Try and copy all the files from the SD card to your PC/Tablet/Phone/Toaster/ect. and format the SD Card. Restore the files afterward to the SD card and re-insert into the system.[br]
This one is NOT recommended,due to it requiring to take the system apart. You would need to disassemble the system (According to what 3DS you have). You would then need to remove the Wi-Fi board and use compressed air or blow it off and re-seat (Put back/replace) the wi-fi board and reassemble the system (Don’t forget the L and R buttons!) and try to power on. You could inspect inside the system for damaged/un attached ribbon cables that require fixing.
Hopefully I helped anyone in need!