In Attic Greek, "rough breathing" (
/h/ at the start of a word) was apparently phonemic: it couldn't be predicted from context, and had to be memorized as part of the word. Byzantine scribes later invented a diacritic for it, which we still use today.
But are there any minimal pairs distinguished by the rough breathing? That is, are there any two words that are completely identical in all respects, except that one has a rough breathing and the other has smooth?