Paul being a descendant of Jonathan would have some appeal from a devotional perspective since Jesus' more direct saving of Paul could be viewed as fulfilling the covenant of friendship between David and Jonathan and their descendants (1 Samuel 20:42).
From Philippians 3:5 we know that he was from the tribe of Benjamin (like Jonathan) and Paul's other name, Saul, might be more common among descendants of King Saul than among Benjaminites generally. On the other hand, with the purging of the house of Saul (2 Samuel 9:3 indicates that Mephibosheth might be the bottleneck as a sole survivor) there might have been few if any descendants of Jonathan in the first century A.D.
Is there any other evidence supporting or falsifying this possibility or is this merely a wild speculation where even tradition is silent?
Optional bonus question: Has this speculation been written about earlier in Church history? (Allegory and other somewhat fanciful conceits seem to have been more popular earlier in Church history, so I would not be surprised if someone had considered this possibility given its devotional attractiveness.)