I've bought a 3DS that when turned on the menu on the bottom screen always registered right button press, this caused the selection cursor to always float right.
The joystick was even replaced but it made no difference.
Cleaning all of the contacts with IPA made no difference
Button dome contacts were cleaned, again there was no difference.
The only way to make it stop was to disconnect the joystick completely.
I disassembled the unit and left the joystick dangling, I noticed that the drift stopped.
At this point I figured out the drift was not caused by the stick itself but by the connector.
I pressed and bent the metal pins a bit to make a better connection and the drift was fixed.
The location of the joystick connector.
Try pressing down the pins or replacing the connector.