Would you expect a person with a resting heart rate in the mid/upper 50s and who works out 5 mornings a week, to routinely have an HRV score in the mid 60s?

Any suggestions regarding next steps to take to increase HRV?

Sure I'd expect that.  I'd also expect that person to routinely have a score in the mid 70s or mid 50s or ... The score is rather personal, and it doesn't make much sense to compare the number between people.  Any given number could be routine for a given person.

More interesting would be to know what happens to a person's HRV after training and rest days.  I'd imagine the person described would see an increase in HRV on the second rest day.  Is that the case?

If you're talking about lasting changes in the average HRV, then theoretically aerobic work is called for.  However, I think I do primarily aerobic work three times a week for a total of about 9 hours and my HRV average has remained arguably the same for the last 5 months.  Meditation is also supposed to increase your HRV.