What methods are used to make a turbofan engine spool up quicker?
Several things come to mind, but they will all reduce efficiency and/or cost money, so they will not be popular with engine users:


For turbofans: Reduce the rotational inertia of the spinning parts by reducing the bypass ratio. Inertia is mass times distance, so making the engine diameter smaller should have the biggest effect.
Increase temperature margins for the turbine. Today, the electronic control (called FADEC for Full Authority Digital Engine Control) will limit the amount of fuel that can be injected into the combustion chamber to limit the temperature in the turbine. By running the turbine hotter for a few seconds the turbine torque can be increased, at the cost of a reduction in the lifetime of the turbine.
For turboprops: Reduce blade pitch momentarily to unload the propeller, so more torque is available for spooling up the engine core. This, however, reduces thrust when it is needed most.
Reduce secondary loads by disconnecting generators and closing bleed valves. This is already done by todays engine control, however.


The best way of dealing with the slow engine response is operational: Avoid situations where the need for a rapid spool-up might arise or enter those situations not with engines set to idle, but to some medium power setting. Carrier pilots do this routinely by spooling up the engine shortly before touchdown, so they will have the required thrust at their disposal if the tail hook fails to catch any of the arrestor cables.

Thanks to RalphJ for pointing out that turboprops use the same tactic. Helicopters never let the engine run in idle while in flight. If less thrust (turboprops) rsp. lift (helicopters) is needed, both adjust pitch to set the desired thrust/lift and keep engine RPM up. Now the mass flow through the engine core is the same as at full thrust, only the amount of heating in the combustion chamber is reduced to adjust the power output of the engine. When more thrust/lift is needed, the pilot adjusts pitch and fuel flow to set the desired thrust/lift level. Since the engine is already running at top speed, the change is not delayed by inertial effects.