The optimum altitude indicated by the FMS differs from what the planning/dispatch charts in the FCOM indicate would be the optimum altitude. For example, the FCOM volume 1 page 369 shows an optimum altitude of 34500 for a weight of 65000 kilograms, but the FMS indicates an optimum altitude of FL377 for this weight. Bug? Normal behavior? PEBKAC?

It's probably also a function of Cost index (just a guess)

Are you looking at the actual optimum/max fields while in flight or are you looking at the one (TRIP or something like that) that's on the PERF INIT page? The one that's on perf init isn't an optimum, it's a minimum cost altitude according to some tables Boeing gave us - a lot of airlines don't even have that field actually.

Same for both, it looks like. Right now, for example my aircraft weighs about 60Kg. The FCOM tables say the optimum altitude is FL377, FMS CRZ page says FL394.

The FCOM covers the non-winglet case. There are differences between winglet / no winglet airframes based on additional documentation.

Thanks