The first one is better. :) You could also say, "Something about him makes me think about my father, but I can't put my finger on what (it is)." Or "Something about him reminds me of my father, but I don't know why." Or "I don't know why, but something about him reminds me of my father."

I would choose the second option. It sounds less eloquent or literary to end the sentence with the word "what."

Should you say "Something about him makes me think of my father, but it isn't clear exactly what" or "Something about him makes me think of my father, but what exactly isn't clear"? Between THESE two, which one is right? Between these two, the first one is right. It's perfect just the way it is, in my opinion. Good job :-)