"It is possible that Paul’s “relative” Lucius is Luke, the author of the Gospel and the Acts of the Apostles. On his second missionary journey, Paul may have gone to Troas (where Luke lived—or at least where he joined Paul) because he knew a relative he could stay with there (Acts 16:8, 11)." - http://www.christianitytoday.com/ch/1995/issue47/4702.html
I know a related question is this: Who was Luke in the Bible? But I find the answer lacking. I recently heard teaching on slavery in the New Testament by Michael Card, and he presented Luke as a slave, and physicians in general as slaves in the time of the New Testament.
The 'doctors' in ancient Rome were not nearly as highly regarded as the doctors in Greece. The profession itself, outside of the legions, was considered a low social position, fit for slaves, freedmen and non-latin citizens, mainly Greeks. unrv.com
He also taught that slaves in that time were named one of two way, 1) by the master's desired characteristic for the slave (the most common slave name, philokurios - loving of the master, and Philemon - dear one), or 2) by a nickname of the master (i.e. Michael would name his slave Mike).
Paul mentions a relative, Lucius. Lucius would name his slave Luke. As a physician, he would have been given to Paul to care for his ailment, (or thorn?). That would explain why Luke traveled so extensively with Paul.
Michael Card does have a book on both slavery and Luke, and I will be getting to those, but how come I've never heard this before? Is there substantial evidence pointing to a definitive yes or no?