Hi,
The original quote was " Someone figured out my password and is posting under my account. How dare them." Now, it should either be 'How dare him', or 'How dare he/she' or 'How dare they'
The question is basically the difference between these two cases.
He stole the cookie. How dare he.
And
He stole the cookie. How dare him.
Now it seems to me that it should be him as it sounds like an objective case, but I really do not know . . .
The subjective form is correct. 'He', 'she' or 'they' if you are not sure about the person's sex.
It's just a matter of an unusual word order. When you say 'How dare he do that?', you are saying 'How does he dare to do that?' You are not saying the ungrammatical 'How does him dare to do that?'
Best wishes, Clive