OED has zilch on this term. Am I right in thinking it’s a combination of GI and Marine? How and when did it arise? Were all Marines called gyrenes or did it refer to only a subset performing some particular function?

GI+Marine sounds logical, but MW dates it to 1894—too soon for GI.

HDAS has it. It dates to 1894 and the first citation is from a glossary included in the first issue of the US Naval Academy’s yearbook, Lucky Bag.

Lighter says the origin is unknown, but suggests it may be taken from the Greek gyrinos “tadpole, pollywog,” a reference to the Marines’ amphibious nature and also a rhyme. (Back in 1894 one could expect midshipmen to have studied some Greek.)

The 1894 dates precludes the GI + Marine explanation, as US Army soldiers weren’t called GIs back then.

Until the Women’s Royal Naval Service was integrated into the Royal Navy in 1993, Wrens serving with Royal Marine divisions were known as Marens. For all I know they still may be; the designation ‘Wren’ has been officially obsolete since integration but is still unofficially used, so it wouldn’t surprise me to learn that ‘Maren’ is too.