My mother lives in an older house.
Some of the outlets are wired with 14/2 w/earth gnd (3 pronged outlets).
Some are wired with 14/2 w/o earth gnd (2 pronged outlets).
My mother uses a lot of "cheater plugs" (adapts 3 pronged cords to 2 pronged outlets).
This leaves the appliance cases ungrounded (no earth protection at all).
One solution is to rewire the house (ugh).
Another solution could be to replace the 2 pronged outlets with 3 pronged outlets and wire the earth ground prong to the neutral of the outlet.
This second solution could place a small voltage on the appliance case due to IR drops in the neutral wire.
What is your opinion of this solution?
Is there a better solution?

Whatever you do, I would NOT wire the grounds of receptacles to the neutrals. I'm pretty sure this is against code and could be dangerous.

This practice is wrong, incorrect, not allowed by Code, as it is a fire and safety hazard:



Another solution could be to replace the 2 pronged outlets with 3 pronged outlets and wire the earth ground prong to the neutral of the outlet.
This second solution could place a small voltage on the appliance case due to IR drops in the neutral wire.

It does not apply voltage, but it does allow current to flow in the grounded surfaces.

To provide an actual equipment ground conductor, the only method is to rewire with a wiring method that has an equipment ground conductor.

You may use a GFI device for added safety from ground faults, but this does not provide a ground, any 3 wire GFI protected receptacles must be labeled "No equipment Ground" these labels are provided with GFI receptacles.