I have an old answer on the popularity of the name Thomas in 16th century England that I think will be useful here!
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Popular names in the later Middle Ages really tended to cluster. Consider that 50% of Henry VIII's wives were named a variety of Catherine and 33% of them Anne. (And Anne Boleyn's father and maternal grandfather, in addition to the uncle you name, were both Thomas).
In some cases it's simple generational transfer--Anne's uncle Thomas Howard is the oldest son of Thomas Howard. The zeitgeist for the popularity of this particular name, however, is almost certainly the cult of 12th century bishop-martyr-saint Thomas Becket. (Latin cultus, devotional practices surrounding a particular saint or Church celebration like prayers of intercession, dedicated sermons and hymns, pilgrimages to a shrine, etc.)
Possibly after St. George (England's patron saint), Becket was the saint of late medieval England. The popularity of martyr-saints towards the end of the Middle Ages (in conjunction with a lack of new actual martyrs) only promoted Becket's popularity. Recall that Chaucer's pilgrims are en route to Becket's shrine at Canterbury. He usurped a place for himself alongside Paul as co-patron saints of London. Parish and civic guilds frequently adopted him as their patron saint. While plays revolving around saints' lives and miracles weren't as common in England, there are actually quite a few examples of Becket ones from the late fifteenth and early sixteenth centuries. They were popular enough among potential audiences that they're noted to serve as fundraising events for some of those guilds! And Becket's murder certainly brought the entertainment. Surviving records note payment to a local laundry for washing Becket's "blood" off the vestments worn by the actor during the martyr scene.
It's interesting to note that in the sixteenth century, the various efforts at a Protestant Reformation went after the cult of Becket with extra zeal--even to the extent of crossing out his name in manuscripts (but save the artwork!), or cutting out just his specific feast day from liturgical calendars. That's more likely a reaction to the actual saint's perceived role as martyr for the Church standing up to the king, than any connection with the multitude of reforming Thomases around Henry.
Pictures courtesy of the amazing British Library's digitized manuscripts collection.
