My hot tub was wired for a 240V line in the previous owners location. Looking at the wiring diagram, and in the control box itself, I see that it can be fitted with a 120V line instead (provided the 120V line is on it's own circuit).

The manual I have is generic so it doesn't go into how to actually do this wiring. After studying the manual, it seems to be a matter of 1) switching the wiring harness inside the control box from 240V to 120V (little white plugs... just switched them) and 2) wiring up a new GFCI protected 120V (and at least 20A) power source to the bare wires in the control box. I've done the first part, but the wiring has me a bit confused. So, here is my question:

These are 3 leads and a grounding screw in the control box. The leads are White (neutral), Black (L1), Blue (L2). My new power source cable has 3 leads... White (neutral), Black (hot), Green (ground.) I connected the Green to the ground screw in the control box, the White to the White in the control box and the Black to the Black in the control box. I wasn't sure what to do with the Blue (L2) so I capped it off for now.

When I tried powering up this setup, my GFCI breaker switched off. Something isn't right. Can you shed any light on this? Should L2 be used in some other way?

Also, before the GFCI switched, I noticed that the "Miswire" light illuminated on the control box. Definitely something not right...

There's a high degree of certainty the answer to your question is dependent on the actual brand of spa you're working with.