I'm not sure if I know of any examples. When two words fall together and can no longer reliably distinguished, then either that single word takes on the meaning of both, or the old meaning is retained and new word is used to cover the gap. I'm not aware of a name for instances where another word was changed to make it less similar to another.
Can you give me an example of when a new word is used to cover the gap?
I see. So, if a phonological change happened to **ē to make it dissimilar, could you guess what it potentially would become?
Phonetic changes almost never take such considerations into account. They are kind of at a lower, more basic level, and are not normally affected by the meanings of words. It does sometimes happen, though, that when sound changes make certain words too similar, a language borrows an alternative form from another dialect or even another language. they was borrowed from Old Norse in that way, to replace the original word which had become too similar to he.