A MAJOR fallacy in the VFD sales game is for people to say that a VFD will ALWAYS save energy. That is blatantly untrue (and yes, I an in the VFD business). A VFD can only save energy on centrifugal loads such as many pumps and fans, but more importantly, it only saves energy IF it is replacing some OTHER means of VARYING* the flow. If the flow never varies, then the VFD will actually INCREASE the losses in the system, thereby WASTING energy.
The reason for the success in REPLACING OTHER FLOW CONTROL METHODS is because of what is called the "Affinity Laws". One of those laws of physics dictates that the power needed by a centrifugal load varies as the cube of the change in speed. So for example if you run the pump at 50% speed to reduce the flow, the power required by the pump from the motor is going to be .5 x .5 x .5 or 12.5%, or rather at 1/2 speed, the pump only demands 1/8 of the power it did at full speed. So because the VFD can exactly match the motor output to the pump demand by changing the speed, the law follows through and reduces energy.
Changing flow by using a throttling valve ALSO reduces energy, but it is not by the cube, it is direct. So the energy savings in using the VFD is in the DIFFERENCE between using the motor speed to vary the flow, or some other less efficient means of varying the flow. However, again, if you do NOT vary the flow, there is NO savings.
* Notice that I use the term "vary", not "change". That's because they are different issues. Varying means constantly changing, but if you just want to PERMANENTLY change the flow in a pump, and it will always still run the SAME speed, then it is more efficient to just have the impellers trimmed. VFDs lose about 3% of the energy through them (as heat). If you gain more efficiency in the variation, that becomes irrelevant. But if you always run at the same speed, you have to compare that to trimming the impellers, and trimming wins. Also, NEVER add a VFD and then run it at full speed all the time because it will be a total waste of money, unless you have a machine with a motor that can ONLY be 3 phase, at which point the VFD represents less losses than any other type of phase converter.
So back to the pool filter pumps. I rarely see a pool filter system with a servo controlled proportional valve that changes the flow, but it happens on larger commercial systems like municipal pools or big hotels. I have NEVER seen that on a residential pool, which by virtue of the 115V and HP size you mention, means tht is what is being discussed here. The only thing I have seen is two speed pumps where the low speed is used most of the time to just run the filter, then once or twice a day they connect a cleaner that uses the high speed. If THAT is what he wants to replace, there may be some merit to it, but it depends on the type of 2 speed motor he uses. If it is a Variable Torque 2 speed motor, probably not going to save energy. How you tell is by looking at the nameplate. If the low speed shows a lower HP, and the HP is lower by more than the speed difference, that is a variable torque version. If it says 2 speeds but only shows one HP, then the VFD will save him energy.