Could you please tell me which one of the following sentences sounds more natural to you? I mean, which preposition sounds better with the word ‘tease’, ‘by’ or ‘with’?
1- He teases me by my father’s name.
2- He teases me with my father’s name.
To start with I think the use of ‘tease’ in your sentences sounds a bit odd - I can’t quite see how ‘father’s name’ fits in. With tease you really need some word to introduce what the teasing is about. Let me give some examples:
He teases me about my appearance/my hair style/my accent. (You could also add ‘my father’s name’.)
He teases me with remarks about my appearance etc. (You could also add ‘my father’s name’.)
Or, and I think we’re now getting to what I take as the meaning of your sentence: