Where does he appear in the Bible, and how many times? (Try Biblehub...)
When was this book written? What was happening to Paul? (Try Biblica, which gives you the helpful introductions to each book from the NIV Study Bible. Then look at the old Jamieson Fausset and Brown commentary, which gives a much more in-depth intro - scroll down to the paragraph which begins "His martyrdom is universally said…")
What did Onesiphorus do for Paul? Look at the key verses in The Message (Bible Gateway) - and a few other translations too if you like…
Paul was in the Mamertine prison at this point. What do we know about the conditions in Paul's day? Google links to lots of articles about it.
What was Nero doing to Christians at this time? Livius.org specializes in articles on ancient history.
What does "Onesiphorus" mean? (The Wikipedia article would do for this…)
(And while you're on the Wikipedia page: was he still alive? (Look at the paragraph in the article which begins "But now…" There's a lot more here but probably too much confusing detail - it's a site where anybody and everybody can weigh in with an opinion, and some are more sane than others!)
How well off was he? Does this verse give a clue?
Does his name tell us anything about his social status? Eerdman's Dictionary of the Bible might help here.
What did he actually do for Paul? Starting at this Biblehub page, work out what it means that he "often refreshed" Paul
What would his motivation have been in helping Paul in this way? (Try the GotQuestions article for hints…)
What can we learn from Onesiphorus? Skim this thoughtful sermon and see what points stick out as important…