I’ve often read of Onesimus as being a runaway slave. Interestingly, S. Scott Bartchy suggests otherwise:
“Not infrequently, knowledgeable slaves left their owner’s control temporarily to hide from an angry owner and wait for tempers to cool, perhaps hoping to find an advocate to intervene on the slave’s behalf. Others took off to visit their mothers (Dig. 188.8.131.52-5). According to Proculus, the foremost Roman jurist in the early first century, such a slave emphatically did not become a fugitivus (Dig. 184.108.40.206). In light of this legal opinion, the fact that Philemon’s slave Onesimus did not take off for parts unknown but rather fled to Paul in prison strongly suggests that it is incorrect to view Onesimus as a runaway slave.”
S. Scott Bartchy, “Slaves and Slavery in the Roman World,” in The World of the New Testament: Cultural, Social, and Historical Contexts, ed. J. B. Green and L. M. McDonald (Grand Rapids: Baker, 2013), 172.