The N64 is mechanically robust, but the joystick is its Achilles heel. The OEM analog stick wears out and develops a loose "dead zone" — fortunately it can be rebuilt or replaced with modern aftermarket modules.
LIKELY CAUSE: Worn POM stick gears and lubrication breakdown inside the controller
LIKELY CAUSE: Dirty Jumper Pak / Expansion Pak contacts
LIKELY CAUSE: Dirty AV cable or failing video DAC