It means "saint", "master", or "lord" in Sindhi and Marathi, from the Dari Persian "sāyæ"
Originally, سایه (sāyeh) was a poetic way to refer to Sufi mystics in Persian. It literally means "shade" with the connotation of "protective" and "influential." Sayeh in of itself is a name among Iranians. With the Mughal conquest of India, it was loaned into many languages. Sindhi Sufis added the suffix ین ईं -ī(n) to the root word to make it the adjective سائیں साईं sāī(n) an honorific meaning "saint", "master", or "lord." In fact, Modern South Asian Sufis often refer to God as "Allāh sāī(n)". In the 19th century, a Sufi ascetic in Shirdi, now located in Maharashtra, India, with no name came to prominence as a preacher of religious tolerance and unity and drew both Muslim and Hindu devotees to him. His devotees referred to him as سائیں بابا साईं बाबा "sāī(n) bābā". Over time, the word became standard Marathi but it was corrupted so that the homorganic nasal was deleted and it became साई "sāī". Today, the name Sai is given to both boys and girls in India in reference to Sai Baba of Shirdi, however it is still relatively rare. It is often followed by a middle name which changes the overall meaning and avoids confusion with other people with the same name. It is most common among the people of Maharashtra, Andhra, Karnataka, Gujarat, and Goa.