If the father is the Gabbai ,how should he call his son up for an Aliyah ? Does the father use his own name for example Yaakov ben Moshe? or Yaakov Bnee?
I have been a gabbai and the way I was trained was יעמוד בני יוסף without mentioning my own name. That is also the way that I have seen other gabbaim do it.
When my sons who are gabbaim mention their children in a mi shebeirach (as an example) they also say 'Bni Ploni' or 'Biti Ploni' without using ben or bas and their names. When they call up their brother, they say 'Ploni ben Avi Mori'. When they call me or reference me or my wife it is 'Avi Mori' or 'Imi Morasi'
UPDATE For Yizkor, the nusach in various siddurim and machzorim is Avi Mori Ploni ben Ploni. I have not seen a write up discussing the difference. You can also see a link in the comments to that nusach when calling up also. My answer was based on how I was taught and how my sons were taught in their Yeshivos.
What I've always seen done was יעמוד בני יוסף בן שמעון where the Gabbai is Shimon.