With a quick glance at historical books we see that Abu Bakr ibn Ali, Umar bin Ali and Usman bin Ali were the children of Laila the daughter of Masoud Thaqafi, Um Habib and Ummul Banin, respectively. When reflecting on the similarity of name between these children and the Khulafa it can be said:
The public mindset at that time was in such a way that these names were not significant and subject to attention and accordingly didn’t posses approving or disapproving connotations. As a matter of fact these names would not remind anyone of the Khulafa and it was through the passing of time and history that the names of these three people were bolded, but back then these names were so common that if Ali (A.S.) or anybody else chose the name Abu Bakr for their child it wasn’t reminiscent of the first Khalifa. Moreover there were great companions of the prophet (A.S.) that had these names like Usman bin Maznun and others.
These names were in that era and, as witnessed in historical books, later on used commonly in the time of the other Imams. Besides these names weren’t hated and condemned by the Shiites as Mu’jam Al-Theqat has listed nearly 60 pages of narrators that had the name Umar or Usman bin Saied which was one of the four special representatives of Imam Zaman (A.S.) in the time of the minor occultation.
When referring to historical documents we find that after the martyrdom of Fatimah Al-Zahra, Amir al Muminin (A.S.) had three children that had the same names as the Khulafa. One of them was Abu Bakr bin Ali bin Abi Talib the son of Laila, the daughter of Masoud Thaqafi. The other child was Usman, whose mother was Ummul Banin (which both of them were martyred in Karbala alongside their brother Imam Hussein (A.S.). The third child was named Umar, the son of Imam Ali.
There are a few points that must be noted regarding the naming of these children:
1. Basically naming your children with same names as somebody else doesn’t prove your affection for them and doesn’t mean you share the same beliefs and fiqhi and political stances, the same way not using those names doesn’t necessarily suggest any sort of friction and enmity, although undoubtedly, love and hate sometimes might play a role in this issue.
In addition, the public mindset at that time was that using those names wasn’t a significant thing and was not conceived as a sign of approval or disapproval. As a matter of fact these names would not remind anyone of the Khulafa and it was actually through the passing of time and history that the name of these three individuals were bolded, but back then these names were so common that if Ali (A.S.) or anybody else chose the name Abu Bakr for their child it wasn’t reminiscent of the First Khalifa.
One must know that the names of the three khalifas were not confined to them, but actually frequently used before and after Islam, among Arabs. This naming could not be a reason for friendship and liking. For example, the last king of Iran, named Muhammad Reza, was the most despised person in the country, but at the same time this didn’t change the fact that this name was commonly used among people. There were others among the great Sahabah (The Companions) of the Prophet (A.S.) that had the same name as the Khulafa too such as Usman bin Maznun and others so how can someone claim that these names weren’t for the remembrance of these great companions. You can check the names of the Sahaba that were named Abu Bakr, Umar and Usman in Al-Isti’aab by Ibn Abdul Birr and Asad Al-Ghabah by Ibn Athir.
Now we will take a brief look at the book <<Asad Al-Gabah Fi Maarifah Al-Sahaba>> and list those who had the name Umar.
1. Umar Al-Aslami, 2. Umar Al-Jami’I, 3. Umar bin Al-Hukm, 4. Umar bin Salim Al-Khazai’i, 5. Umar bin Siraqah, 6. Umar bin Sa’ad Al-Anmari, 7. Umar bin Sa’ad Al-Salma, 8. Umar bin Sufyan, 9. Umar bin Abi Salmah, 10. Umar bin Amir Al-Salma, 11. Umar bin Ubaidullah, 12. Umar bin Akramah, 13. Umar bin Amr Al-Laisi, 14. Umar bin Umair, 15. Umar bin Auf, 16. Umar bin Azyah, 17. Umar bin Laheq, 18. Umar bin Malik bin Uqbah, 19. Umar bin Malik Al-Ansari, 20. Umar bin Muawiyah Al-Qazeri, 21. Umar bin Yazid, 22. Umar bin Al-Yamani.
These are just the people whom Ibn Asir has mentioned in his book.
We can clearly deduce that these names have been commonly used throughout history, both before and after Islam and frequently heard in all groups and classes.
Also, there are many of the Tabi’in (The Followers) that had the name Umar, the book Mu’jam Al-Thiqat has listed nearly 60 pages of narrators that had the name Umar.
These names were very common at the time of the infallibles such as Abu Bakr bin Abdullah Sa’d Ash’ari and Abu Bakr Hadhrami who were narrators of Imam Sadiq’s (A.S.) hadith, and Usman bin Saied which was one of the four special representatives of Imam Zaman (A.S.) in the time of the minor occultation. All of these facts prove that using names like Abu Bakr, Umar and Usman would not remind anyone of anybody except the owner of the name himself. It isn't as if there was only one Ali and one Abu Bakr (Umar or Usman) and despite this scenario, Imam Ali still went ahead and named his children Abu Bakr, Umar or Usman; these names were commonly used by different people at various times.
(Of course, if naming a certain name is ever ordered by God or His Prophet it will carry with itself a sacredness, as is the case with the names of the children of Ali (A.S.) and Fatimah (S.A.) children; Hassan and Hussein (A.S.).)
Therefore, you will agree that the names of the other Khulafa were famous and common names before and after Islam and no one would ever remember the Khulafa when hearing them. Therefore we cannot overlook all of what has been written in history regarding the enmity Imam Ali had with the other Khulafa because of a simple similarity in name.
2. The Imam, with his high wisdom and insight, would hide his pains and discontent and would take part in helping the Khulafa for the sake and interest of the newly emerged Islam, to the point where Umar stated on several occasions: «لو لا علی لهلک عمر»  (If it wasn’t for Ali, Umar would have certainly perished). Some have the theory that these names were used to compromise with the Khulafa, for this would reduce the friction and tension in the Muslim society, not that it was an approval and sign of truce and friendship with the mentioned individuals. In any case, if we consider the suppression and pressure imposed on Shiites at that time, we will conclude that in order to protect the Shiites the infallibles were sometimes compelled to act (taqiyah) in a different and maybe controversial, but, still, Islamically permissible way, like: naming their children with the names of the Khulafa and creating ties through marriage with the great Sahabah so that the oppressive government of the Bani Umayyah and Bani Abbas would not persecute Shiites even more by stimulating ignorant people to ravage and kill them with the excuse that they are against the three Khalifas.”
It is useful to note that nowhere in our historical books (that have been studied) has it been mentioned that Imam Ali directly chose these names, hence there is a possibility that these names were picked by his respected spouses or others which the Imam respected and wouldn’t argue with their decisions.
 Mu’jam al-Thuqat, vol. 21, pg. 66.
 Al-Irshad, pg. 484.
 Mu’jam al-Thuqat, vol. 13, pg. 45.
 Muntahal-Amal, vol. 1, pg. 220.
 Tabari, Muhammad, Pasokhe Javane Shi’i, pp. 55-56.
 فَرَأَیْتُ أَنَّ الصَّبْرَ عَلَى هَاتَا أَحْجَى فَصَبَرْتُ وَ فِی الْعَیْنِ قَذًى وَ فِی الْحَلْقِ شَجًا أَرَى تُرَاثِی نَهْبا Nahjul-Balaghah (Sobhi Saleh), pg. 48, sermon 3.
 Tabari, Muhammad, Pasokhe Javane Shi’i, pp. 56-57.