I have been looking at the halachot for a son who is the gabbai and who calls his father up for an aliyah here and I know that there was a question about it, but my question is about after the aliyah. When making the מי שבירך afterwards, one often includes the name of relatives. How does the gabbai refer to himself if the father wishes to give a bracha to his children? The gabbai adds in something like "avi umori" out of respect, so he probably doesn't need to add "achi" if his brother and family get the מי שבירך but does he refer to himself and his children in the third person?
I have been a gabbai and my sons are gabbaim. My sons refer to their brothers as 'Ploni ben Avi Mori' and their sons as 'Bni Ploni' without using their own names. I think that they would also refer to their wives as 'Ishti Plonis bas Ploni' and that is often how I would do it when I was gabbai.
Note that we would first check with the rav of the shul to make sure what the minhag was before doing this. This was not just something that we came up with on our own.