When will the next series of mutual eclipses of Jupiter's moons begin?
Mutual eclipses and occulations of Jupiter's moons occur during Jovian Equinox, twice in each orbit of Jupiter, and so roughly every 6 years.

The last series of events was in 2015, so the next will start in 2021. There will be a series of events between January and August. 

The site http://lnfm1.sai.msu.ru/neb/nss/nsszph517he.htm can generate ephemerides for these events.