Bill is probably short for Billie, which is the modern feminine form of William. Wilhelmina is very rare nowadays. Might also be a shout out to Billie Piper, the first new companion. The companions can be interpreted as the Doctor's substitute family but this has been hammered home far more obviously for Bill than any other companion. The reference to Billie Piper (the original for all new Who watchers) thus also echoes the first old companion, Susan.

Given that Bill combines the themes of agency and family, her name being Will and Protection fits both parts very well.

Potts should perhaps be interpreted in the obvious British way - Potty. A bit mad.