The biggest source of homonyms are words imported from Chinese, as Chinese morphemes are usually monosyllabic. It is already a problem in Chinese due to the limited phonotactics, made even worse in Japanese.

So the most obvious solution would be to drop on'yomi (Chinese readings) and go to pure kun'yomi (Japanese readings) whenever possible. My understanding is that such a strategy was used by the Koreans to replace Hanja with Hangul.

Now, I understand that it would be a massive undertaking and extremely unlikely to ever happen, and honestly it's not really my problem, so I am just speculating here xD