The verb ἵημι is notoriously fertile ground for such minimal pairs, when compared with εἰμί and εἶμι, e.g.:
εἷναι aor. act. inf. / εἶναι pres. act. inf. of εἶμι
ἱέναι pres. act. inf. / ἰέναι pres. act. inf. of εἶμι
ὧ pres. act. subj. / ὦ pres. act. subj. of εἰμί (in all persons/numbers)
εἵην pres. act. opt. / εἴην opt. act. subj. of εἰμί (in all persons/...
Here's a list I was able to generate from the Perseus lemma list. This only looks at headwords, so it might exclude a few words that have similarly declined forms. Some of the words are also a bit non-standard, but the LSJ has them all listed:
ἅλινος - relating to salt
ἄλινος - without a net
ἀνία - grief
ἁνία - reins
ἐδανός - eatable
ἑδανός - sweet
Since some time has gone by without anyone else supplying documentary evidence concerning the matter, I'll provide an answer based on what I think must have happened, but without any ancient quotations to back it up.
First of all, the name of the letter Η must have been heta (ͱῆτα or hε͂τα) at the time when the letter was still being used to indicate /h/ - ...
I'm not sure whether you meant for it to go without saying, but here are some basic facts about the distribution of the rough and smooth breathing marks in polytonic Greek orthography. The rough breathing is thought to have represented aspiration (possibly a consonant phoneme /h/, but there are more complicated suggestions for its phonemic representation) ...
I've long relied on the free Gentium typeface, which has a simply gorgeous, highly readable polytonic Greek font, with diacriticals that I've always found quite easy to distinguish, both on the printed page and on the computer screen. I used to have a custom stylesheet to display all Greek text on the Perseus website in Gentium, because it's much easier on ...
A few more:
ὅσσα "as much as" (poetic form of ὅσος as neuter nominative/accusative)
ἐνί "in" (poetic form of ἐν)
ἑνί "in one" (dative masculine and neuter of εἷς)
ἥ "which" (feminine of ὅς)
This tells us that in Hippocrates, Plato, and elsewhere, the name of the letter is spelled with smooth breathing. (Of course the breathings would have been added by Alexandrian editors, so we can't be 100% that Plato didn't actually call it ἧτα.) The last citation, with rough breathing, is your scholium on Dionysius Thrax.