Note that I do not say "in the Eucharistic Prayer".
Because the EPs of other rites and the newer "Roman" EPs may have a different theology from that of the Canon Romanus.
More than half a century ago, Dom Eizenhofer (Sacris Erudiri 1956, 75 gives the Latin summary) demonstrated, in my view conclusively, that the word "Communicantes" goes grammatically and theologically with the end of the Te igitur (Memento being an originally diaconal parenthesis). The grammar is "una cum ... communicantes". And that the theology of the Prayer means that our sacrifice is commended to the Father as acceptable because we are offering it in and for the Church in union with its [earthly] head the Bishop of Rome. He backs this up with a great many pieces of contemporary Latin showing that the language expresses the ideology of the Roman See at the time the Canon acquired its present state: that being in communion with the Roman See is the touchstone of Catholic communion.
Of course not everybody accepts that notion. But what Eizenhofer's demonstration makes clear is that it would not be proper to substitute another prelate for the Roman Pontiff unless one were prepared at the same time to argue that he is not just a Catholic bishop, not just the Head of a Communion, but the actual Prelate communion with whom gurantees one's Catholicity.
So the old Anglo-Catholic ploy of naming the Archbishop of Canterbury, or the Orthodox 'Western Rite' practice of naming a Patriarch, are improper unless one really does believe that communion with that prelate is the universal touchstone of whether anybody is in full commuion with the Church Catholic.
It is also worth noting that this Naming is not a prayer for the Roman Bishop. It is true that we should pray for him; but that is not what we are doing here.
We are expressing our Communio with the Successor of S Peter. I think it was S John Henry Newman who spoke about the Soliditas Petri ... In communion with him, we know that we are in the One Fold of the Redeemer (Newman again).
I find it easy, during this disastrous pontificate, to sympathise with those who question whether Bergoglio really can be pope. But you and I cannot decide such matters. The constitution of the Church is not a DIY game for anybody to play. Our duty is to bear witness against the gross errors of this pontificate, while at the same time allowing nobody to prise our fingers away from Communio with the Successor of S Peter.
That is why, every morning, I name Francis in the Canon of the Mass; and why, dear reader, your faithful priest does the same.