Can magnets break LED notification light on phone?
Taking it as a design question, yes, it is possible.

In a typical design, the LED is driven by an output port on a CPU, microcontroller, or via a FET or transistor.  A loop of conductor runs to the LED and returns.  The return path may not follow the outbound path.  A rapidly changing and strong magnetic field can induce current into the PFC or PCB trace that connects to the LED.  This current may do damage to the driving circuit or the LED.

However, it would have to be a quite significant magnetic field to generate enough current for enough time to do damage.

I'm talking like the sort of fields found in an nMRI machine.  I've never heard of handheld magnets doing this.  It seems more likely that the failure was caused by ESD or physical force.