Why the need to Invoke the Demon's Nemesis (Saint) in the Rite of Exorcism?

Often, I get a good (and often standard) question about how things work in the realm of spiritual warfare.  While they often have to do with exorcisms, the work of an exorcist in casting out demons is very much related to our own spiritual work in resisting diabolical attacks, whether ordinary or extraordinary.
Q:  Why does the exorcist need to invoke the name of the demon's nemesis (Saint) if all you need is the Holy Name of Jesus to cast the demon out?  And why would the exorcist want to torture the demon (which is what happens) by calling upon the nemesis when he can simply cast the demon out?
Good questions.
The Rite of Exorcism and casting out demons is a complicated affair.  I would say that is the answer.

The Rite of Exorcism is a sacramental and, thus, it is not efficacious simply by the pronunciation of the words, as compared to the words of Consecration in the Holy Sacrifice of the Mass, which are efficacious.  When the priest says, "Hoc est enim corpus Meum," the bread is transubstantiated at that moment into the Body of Christ.

When the priest commands, "Begone Satan!"(or similar words) in the Rite of Exorcism, the demon does not, by those words, have to leave.  It is a process and many factors are in play:
- the strength of the demon
- the number of demons
- the reason the person is possessed (gravity of the sin, cooperation with the devil, a curse, etc.)
- the willingness of the possessed to be liberated
- the holiness of the priest
- the spiritual preparation of the priest and possessed for this particular session
- the prudence and humility of the priest in using the Rite properly 
- the spiritual strength of the Church on earth at the moment
- etc.

The Rite of Exorcism, as a sacramental, relies on lots of additional sacramentals, and the Communion of Saints, to be effective.  It also relies on other powers outside of itself, such as the power of the Holy Names of Jesus and Mary, the Saints, the holiness of the priest and possessed, and the reliance upon prayer and fasting and the Sacraments before the session, to name a few.

The Rite of Exorcism guides the priest to invoke all of Heaven to come to assist the possessed.  It is not simply the invoking of the Name of Jesus that will cast out a demon.  Often, demons will not listen or will mock and ridicule the priest.  All the while, though, the demon's resolve is being eroded.

What is happening in the Rite of Exorcism is the punishment of the demon.  The Rite afflicts the demon.  
Imagine the demon as a tick, who has been discovered only after having dug his claws firmly into the skin and sunk his mouth into the flesh.  It takes a careful effort to remove the tick at that point.  If you had noticed the tick crawling on your arm before it latched on, a simple swat with the finger would have been sufficient.

Likewise, when someone is possessed, the demon, who first appeared as a temptation, a nuisance, an intruder, etc., is now in the position of "invader" and "violator" and "latched-on tick."  
If you notice the demon approaching you as a mere temptation or lesser attack, then the Holy Name of Jesus would be sufficient to drive it away.  Even then, depending on the state of the soul of the person involved, there may be a need for immediate Confession, Holy Communion, the use of holy water, sacred medals, house blessing, etc., to drive that external demon away for good.

Once you let the demon in, through possession, it is not so simple to drive him out - nor should it be!  
The journey toward liberation is both a punishment to the person and a sure means to sanctify his soul.  It is medicinal, and necessary.

To rid the possessed of the demon, the priest must break down the demon's hold on the person.  This is a process of spiritually pummeling the demon with all of the sacred invocations the priest can muster!  The Holy Name of Jesus is sufficient, but not so much so that the mere utterance of it will resolve all issues.

As we know, it is the holy ones, the Saints, who perform miracles.  Average run-of-the-mill Catholics, with lots of spiritual baggage and weakness, do not perform miracles.  God is able to work extraordinarily in the ones whose souls love Him extraordinarily and have achieved an extraordinary purity and holiness.

As a result, the Saints are able, from Heaven, to share the merits of their holiness, in union with Christ, in the process of the Rite of Exorcism.  This points to the spiritual war, not only between man and Satan, but also between the Communion of Saints (and Angels) and the Kingdom of Satan.

We are all fighting, and Christ is fighting in and with and through and for us.
So, the Holy Name of Jesus, wielded by a priest, is not sufficient to cast out the demon.  The deficiency lies not in the Holy Name of Jesus but in the one who wields it.  
"If you had faith as a grain of mustard seed, you could say to this sycamine tree, ‘Be rooted up, and be planted in the sea,’ and it would obey you." (Luke 17:6)  Our Lord also says, "This kind [of demon] cannot be driven out by anything but prayer and fasting." (Mark 9:29)  Further, Our Lord says, "O men of little faith," to the Apostles, (Matthew 8:26) which also applies to us all, including exorcists.
An exorcist stands as Christ to the demon and opens up against the demon the torrent of grace which Our Lord has imparted to His Church.  
This torrent must flow through the channels of grace which Our Lord instituted.  
These channels, be they the Holy Name, Our Lady and the Saints, the multitudinous sacramentals, or prayer and fasting, are essential weapons in our efforts, with Christ, to trample down the ancient Enemy of our souls.