This is a transitional verse. The NASB joins it to John 20:1-9 whereas the NIV connects it with John 20:11-18. Since John 20:11 begins with "but," it seems natural to view John 20:10 as beginning a new paragraph. [source][source][source]
The translation "to their homes" implies that Peter and John had permanent residences in Jerusalem. That seems unlikely. The Greek phrase eis ta idia literally means "to their own" (cf. John 1:11). Since the gender is neuter, John may have meant that these disciples returned to their own friends or temporary lodgings (cf. Acts 12:12). [source][source][source]
Context Summary
John 20:1-10 - The Witness Of The Empty Tomb
In considering John's account of the Resurrection, we should remember that it is largely supplementary to the other narratives. This Gospel having been written long after those were in circulation, the selection of incidents which are recorded is made for spiritual purposes. John's object was to show various instances of faith in the risen Christ, each one being typical and having its own lessons to teach.
Easter morning! What dismay there is in Mary's voice and what consternation in her face! What a mistake also she made, for who can take our Lord away from hearts where He is enshrined! The Greek word used to describe the disposition of the clothes is very remarkable. It conveys the idea that they had fallen together, as if that which they had covered had been suddenly withdrawn.
How much those two disciples missed! Had they only waited, they might have seen the Lord. Do not hurry with wanton haste from the mysteries of our Lord's grave; but learn that on the one hand He was declared to be God's Son, Romans 1:4, and on the other we are taught the victory of faith even over death, John 11:26. [source]
Chapter Summary: John 20
1Mary comes to the tomb; 3so do Peter and John, ignorant of the resurrection 11Jesus appears to Mary Magdalene, 19and to his disciples 24The incredulity and confession of Thomas 30The Scripture is sufficient to salvation
Greek Commentary for John 20:10
Unto their own home [προς αυτους] “To themselves.” Luke (Luke 24:12) has προς αυτον pros hauton about Peter (“to his home”). This use of the reflective pronoun for home (literally, “to themselves”), like the French chez eux, occurs in Josephus (Ant. VII. 4, 6). John had taken the mother of Jesus to his home (John 19:27) and so he now hurried home to tell her the glorious news as he believed. [source]
What do the individual words in John 20:10 mean?
Went awaythereforeagaintotheir [homes]thedisciples
Greek Commentary for John 20:10
“To themselves.” Luke (Luke 24:12) has προς αυτον pros hauton about Peter (“to his home”). This use of the reflective pronoun for home (literally, “to themselves”), like the French chez eux, occurs in Josephus (Ant. VII. 4, 6). John had taken the mother of Jesus to his home (John 19:27) and so he now hurried home to tell her the glorious news as he believed. [source]