Basically, he meant something that infects you.
Miasms were not invented by Hahnemann. During Hahnemann's time, there was a debate raging about the origins of disease. There was a miasma theory and the germ theory, which eventually was proved by microbiology, as we all know. "In miasma theory, diseases were caused by the presence in the air of a miasma, a poisonous vapour in which were suspended particles of decaying matter that was characterised by its foul smell. The theory originated in the Middle Ages and endured for several centuries. That a killer disease like malaria is so named - from the Italian mala ‘bad’ and aria ‘air’ - is evidence of its suspected miasmic origins" (Science Museum's History of Medicine ).
Hahnemann was not convinced miasma existed in the air however. Hahnemann's essay "Appeal to thinking philanthropists respecting the mode of propagation of the asiatic cholera" argues that cholera spreads from town to town via minute living creatures.
"In those [people] confined spaces [ships], filled with moldy vapors, the cholera miasm find a favorable element for its multiplication and grows into an enormously increased brood of those excessively minute, invisible, living creatures so inimical to human life, of which the contagious matter of the cholera most probably consists....this concentrated aggravated miasm kills several of the crew; the others, however, being frequently exposed to the danger of infection and thus gradually habituated to it at length become fortified against it and no longer liable to be infected. These individuals, apparently in good health, go ashore and are received by the inhabitants without hesitation into their cottages, and...those who have approached nearest to them are suddenly carried off by the cholera. The cause of this is undoubtedly the invisible cloud...composed of probably millions of those miasmatic animated beings" (Hahnemann's Lesser Writings, page 758) Italics mine.
This passage clearly describes almost the exact same process that happens in infectious diseases, now proven by microbiology. From this passage, his use of the word "miasm" clearly means an infectious disease agent like a virus or bacteria. Therefore his theory on chronic disease does not mean he was thinking about our genetic or epigenetic lineage (those are not infectious diseases) nor does it mean you treat the chronic disease with the infectious disease the caused the miasm (e.g. treat a tubercular miasm with tuberculinum, because a tubercular miasm is the same as an infection by the Mycobacterium tuberculosis bacteria) because that would be isopathy, not homeopathy. He thought that chronic disease was caused by an infectious disease agent that created subtle disease symptoms over decades that would get worse slowly (think of Herpes Zoster/Shingles - the virus infects us when we are children but lays dormant for decades and can manifest in many different ways when our system is stressed).
To me, this is a fascinating example of how well Hahnemann understood the disease process. In other words, that he was a keen observer of the world around him. This remarkable insight was well ahead of his time, and is a testament to the power of being an unprejudiced observer.