As for lunar distance, I suspect the math is not really all that hard. Just like modern celestial navigation there are zillions of tables where all you have to do is reference several numbers in several tables, add them up in various ways, and use the results to references several more numbers in several more tables. Before mechanical calculators (probably even up to the age of electronics) all practical math was done by looking up numbers in tables, and addition.
The reason chronometers are preferable is that the tables procedures are still time-consuming and error-prone (look up the wrong number, copy the number wrong, add the number wrong), so any means you can contrive to remove table lookups is a good idea.
As for what to show your captain, I'd agree with folks who said dacron but I'd go further and say plastics in general, especially fiberglass. Imagine not having to reapply oakum anymore. As for electronics-type stuff, while maps and Google Earth would be impressive, they did already have maps and did know the Earth was round, so the witchcraft involved in animating it is just that -- animation. Might as well show him "the talkies". I would turn on the radio and give him a few minutes to find the man hiding the bilges and failing to accurately predict the weather.
s/v Laelia - 1978 Pearson 365 ketch
s/v Essorant - 1972 Catalina 27