Why does a mouse need a crystal?
That MOS 5717 thing is most likely a microcontroller or some part that executes code.  It needs a clock to run.

However, just a clock for a micro doesn't need crystal accuracy.  That is probably for communication.  USB requires a fairly high-accuracy clock.

A mouse doesn't need to track real time, and there is no reason for it to be in sync with the CPU clock.  Its USB clock must only be close enough to the host's USB clock for communication to work.  Timing about how fast mouse events are occurring or time between mouse events is handled in the host.  The mouse just sends info about what it senses happening.

Added

The above was written in response to the original question, which made no mention of this mouse not being USB.  Since pretty much all new mice have been USB for a decade or more, it was reasonable to answer in that context.  When you ask about something unusual, it's your responsibility to make that clear.

Despite not having USB, this mouse still had a processor that needed to be clocked.  It also apparently used timing to measure the positions of pots connected to a joystick, something else the OP failed to mention.  It seems now that a comment by supercat is most relevant, so I am copying it into the answer:

The Commodore 64 has potentiometer inputs that measure the time required to charge fixed capacitors through variable resistances. Software expects that a mouse will read as a resistance value in the range 0-255, and that it will wrap cleanly 254, 255, 0, 1, etc. which means the mouse has to accurately time its output pulses to within less than 0.4%