The Oxford English Dictionary (OED) attributes genome to Hans Winkler, 1920; the full reference is his book Verbreitung und Ursache der Parthenogenesis im Pflanzen- und Tierreiche, (Verlag Fischer, Jena). At page 165, he writes (in rough translation): "I propose the expression Genom for the haploid chromosome set, which, together with the pertinent protoplasm, specifies the material foundations of the species ...." He discusses this in the context of hybrids that may comprise distinctive genomes from the respective parents, and are then heterogenomatisch. The term was used sporadically in the 1920s and 1930s--Theodosius Dobzhansky scorned it; he would have preferred a "non-committal expression like 'set of chromosomes.'" (1937--Genetics and the Origin of Species)
The OED also offers an etymology, that Winkler's Genom is an irregular formation from gen + some--from chromosome--and this is recopied in many other sources. OED scholarship can rarely be contested, but it has to be challenged here: ...