Possible reason #2 - Christ was praying for specific blessings upon those who were His; obviously He wouldn't do X-and-such spiritually blessing to those who don't believe.
This actually seems pretty plausible. There any many blessings (and salvation!) that are offered only to believers. Paul, in fact, here in Ephesians, prays specifically for blessings upon believers only:
“For this reason I bow my knees to the Father of our Lord Jesus Christ, from whom the whole family in heaven and earth is named, that He would grant you, according to the riches of His glory, to be strengthened with might through His Spirit in the inner man, that Christ may dwell in your hearts through faith; that you, being rooted and grounded in love, may be able to comprehend with all the saints what is the width and length and depth and height— to know the love of Christ which passes knowledge; that you may be filled with all the fullness of God.” (Eph. 3:14-19)
Yet, this does not mean or imply that Paul did not also care about and pray for non-believers at other times:
“Therefore I exhort first of all that supplications, prayers, intercessions, and giving of thanks be made for all men, for kings and all who are in authority, that we may lead a quiet and peaceable life in all godliness and reverence. For this is good and acceptable in the sight of God our Savior...” (1 Tim. 2:1-3)
In His prayer for the blessings of joy and sanctification, Jesus specifically says that He is praying for His disciples and not for the world. Later, though, He prays that the world might believe that He is from the Father; He did pray for the whole world, then, just not in the context of those specific blessings. How could a non-believer be sanctified, or experience unity with God? Those blessings are reserved for believers.
Possible reason #3 - The Lamb of God has an extra special love for those who are His, and still cares about those who He was not praying for at the time.
Given that the church is the bride of Christ, it is certain that He has a special love for them. All others will be cast into hell, but those who believe in Christ are one with Him, and He blesses them richly in this life and the next, working together all things for good for them. Special love:
But one can have a specific love for one, while still caring for others. I have a special love for my husband, but I still agape-love all other men, and would rescue one if I saw him dying and was able to save Him. As my little brother said, on this topic, “Just because I send a letter to my family (special love) does not mean I don't care about everyone else.”
Paul pronounced and prayed for many special blessings solely on believers, and yet also cared and prayed for everyone. Does God also care for everyone? According to the Bible, yes.
“For this is good and acceptable in the sight of God our Savior, who desires all men to be saved and to come to the knowledge of the truth.” (1 Timothy 2:3-4)
So reason number #3 is also Scripturally plausible. Christ not only reserves specific blessings for those who believe, but also has a special love for His own.
Possible reason #4 - Jesus was about to die, and was looking forward to the joy set before Him: His own glorification and the salvation, oneness with, and blessing upon those who would benefit from the cross: the elect, the bride of Christ.
Now, most of the other possible reasons don't look specifically at the timing of this prayer. Jesus was sweating drops of blood that night. The next day, He would experience an excruciating death, the weight of the sin of the world, and separation from His beloved Father. How did He get through this time? This mystery is revealed for us in Hebrews:
“...looking unto Jesus, the author and finisher of our faith, who for the joy that was set before Him endured the cross, despising the shame, and has sat down at the right hand of the throne of God.” (Hebrews 12:2)
Now, if Jesus died to provide salvation for all, and only to procure salvation for the elect (just like the serpent in the wilderness which provided healing for all who were bitten, but only enacted healing toward those who looked at all, see John 3), in which part is the joy found?
Is the joy the glorification and the unity, joy, and sanctification of those who would benefit from the cross? Or is the joy found in those who would reject the provision and wind up justly in hell?
I would say, without a doubt, that the joy would be found (in addition to His glorification) in His bride, in the ones who would benefit and be saved. The joy would not be found in those who reject the work and die eternally – there is justice there, but not joy. Jesus prayed for the world to believe, but that was one statement. The majority of the chapter was focused on Christ's glorification in general, and on blessings for and unity with His elect. That was His focus.
Therefore, it makes perfect sense that as Jesus approached His death, He focused on and looked forward to the JOY set before Him, which was not found in those who would never believe, but would be found in His multi-fold glorification, in the joy and sanctification of His disciples, and of the love and unity between Him and the Father and all future believers – the bride of Christ.
There is the joy that He looked forward to, as He approached the cross. This reason is more than plausible, it is confirmed by Hebrews, which explains Christ's focus on that difficult night.
And if two reasons are found to be plausible, and one is found to be confirmed, then the claim that the first reason is the “only possible” reason must be entirely false.