A lot of people object to "he or she" on grounds of euphony; but clarity of meaning should always take priority in our considerations over sound. The fact is that "he or she" is what we actually mean.

Granted, like any phrase, it is inelegant in certain contexts, and can become tiresome if repeated. So one has to use workarounds. Luckily, "they" (always perfectly acceptable in spoken conversation) is also available for judicious written use.

"the most important reason to argue with someone is to change his or her mind" sounds just fine. ("Their" could also be substituted.)

"Either a person has enough of the rationalist virtues to overcome it, or he or she doesn't" is bad, mainly because of the "or" preceding "he or she". "He/she doesn't" is better, but "they don't" is probably the best (certainly in a comment; maybe a post should be more formal?).

Invented pronouns are just too strange and should be avoided.