How to minimize DC brushed motor arcing noise propagation from MOSFET drain to gate?
Update

Upon reviewing datasheets, the FET has adequate internal diode and looking at the scope spikes, I suggest examining if the ground signal has similar spikes , it is measurement error from excessive long inductive ground clip path.(> 1cm) 

If ground is made clean with clipless probing to ground with 2 pins &lt;1cm apart, then repeat the same type measurement across 5 to 0V . If noisy then add a low ESR bulk tantalum cap and if not quiet &lt;5% ripple and more, lower ESR caps until low noise and then repeat the same test method (clipless tip&amp;ring) on above results and if not resolved, show all captures of gnd, Vcc, Vgs and Vds.

=== 1st cut

The 555 timer has a push pull output which is much lower impedance then the gate. Although there is a Miller capacitance between drain and gate there can be some transient Vgs.      

But the pulse with suggests that the reverse recovery time and ESR of the clamp diode is too high causing twice the supply voltage to appear on the motor during turn-off as we expect with a low side switch.  

The diode is supposed to act as a similar ESR high speed current rectifier to BEMF or high side switch to V+. 

Thus re-examine the choice of diode for similar current ratings with FET and thus greater body capacitance to FET yet short Trr recovery time.  This is normal to absorb the motor stored energy as it gets switched off.

Also , include links to datasheets of all parts in question, to find motor DCR , diode ESR, FET RdsOn and supply ESR of bulk capacitor.