Did the Prophet (blessings and peace of Allah be upon him) have a Jewish servant boy?

Yes, it is proven in Saheeh al-Bukhaari and elsewhere that a Jewish boy used to serve the Prophet (blessings and peace of Allah be upon him). On one occasion he fell sick and the Prophet (blessings and peace of Allah be upon him) visited him and invited him to Islam. This is the story as it was narrated by Anas ibn Maalik (may Allah be pleased with him): 

A Jewish boy used to serve the Prophet (blessings and peace of Allah be upon him) and he fell sick. The Prophet (blessings and peace of Allah be upon him) came to visit him. He sat by his head and said: “Become Muslim.” The boy looked at his father, who was also by his head, and his father said to him: Obey Abu’l-Qaasim (blessings and peace of Allah be upon him). So he became Muslim, and the Prophet (blessings and peace of Allah be upon him) left, saying, “Praise be to Allah Who have saved him from the Fire.”

Narrated by al-Bukhaari (1356). 

But the sources do not tell us the name of this boy or anything about his service to the Messenger of Allah (blessings and peace of Allah be upon him), and they do not mention anything about the reason why the Prophet (blessings and peace of Allah be upon him) accepted his service. 

Al-Haafiz Ibn Hajar said in Fath al-Baari (3/221): 

“A Jewish boy used to serve…” – I have not come across anything  narrated via complete isnaads about his name, except that Ibn Bashkuwaal stated that the author of al-‘Atabiyyah narrated from Ziyaad Shaytoon that this boy’s name was ‘Abd al-Quddoos. He said: This is an odd report; I did not find it with anyone else. End quote. 

All that there is, is the fact that some of the commentators suggested that this boy participated in bewitching the Prophet (blessings and peace of Allah be upon him), as he was the one who took some of the hair of the Prophet (blessings and peace of Allah be upon him) and gave it to Labeed ibn al-A‘sam, the Jewish man who was a practitioner of witchcraft. However this report has not been proven to be sound. 

Al-Qurtubi said in al-Jaami‘ li Ahkaam al-Qur’an (20/232): 

Al-Qushayri stated in his Tafseer that it was narrated in the Saheehs that a Jewish boy used to serve the Prophet (blessings and peace of Allah be upon him), and the Jews contacted him and kept on at him until he took some hairs that had fallen from the head of the Prophet (blessings and peace of Allah be upon him) when he combed his hair, and he took some of the teeth of his comb, and gave them to the Jews, who used them to bewitch him, and the one who did that was the Jew Labeed ibn al-A‘sam. End quote. 

See a similar report in Zaad al-Maseer by Ibn al-Jawzi (9/270) 

From the story of the Jewish boy serving the Prophet (blessings and peace of Allah be upon him) we see an example of the tolerance, easy-going nature and gentleness that the Prophet (blessings and peace of Allah be upon him) carried in his compassionate heart, and that he was merciful and compassionate towards all people; he hoped for good for them and warned them against evil. So he did not hesitate to visit this Jewish boy at home when he was sick, and he did not miss any opportunity to call them to true guidance. But they responded with plots and treachery, and they tried to kill him and put poison in his food. 

For more information on the way of the Prophet (blessings and peace of Allah be upon him) in his interactions with the Jews, please see question no. 84308

And Allah knows best.