In the cathedral shop nearby, we found a cassocked priest to whom we told our concerns. Off he strode, mumbling something uncomplimentary about the morning shift and, when we returned to the shrine some minutes later, the flame was strong and bright.
Of course, if the candle had gone out, it would have been just a minor bump in 1000 years of turbulence for Becket.
Friend and chancellor to Henry II, and on the basis of that friendship named Archbishop of Canterbury in 1162, he suddenly became a ferocious defender of the church's prerogatives - not exactly what Henry had expected.
This famously led to a bitter falling out and, in 1170, to the king's plaintive cry: "What a parcel of fools and dastards have I nourished in my house that not one of them will avenge me of this one upstart clerk!" - or words to that effect - prompting four of his loyal knights to ride to Canterbury and hack the archbishop to death in his own cathedral.
The rain poured down as we followed in Becket's footsteps, from the site of his palace across the grand old cloisters and through a heavy door iron-bound into the cathedral.
There the spot where he was slain is marked by a plain stone altar, where visitors can kneel to offer prayers in his memory, and a dramatic sculpture depicting the swords that hacked him down.
From there, we climbed down the stairs to the crypt, the oldest surviving part of the cathedral, with wall paintings and carvings dating back to the 10th century and, in the centre, the spot where Becket was first buried.
Pilgrims began arriving at the tomb almost immediately, within days the first miracles were reported, in just two years Becket was declared a saint and, before long, Canterbury was rivalling Rome and Jerusalem as a centre for pilgrimage.
This brought the town such prosperity that, in 1220, the shrine was moved up into the cathedral proper, Becket's body laid in a new Trinity Chapel and his shattered skull stored in a golden casket in the adjoining Corona Chapel, whose stained glass windows depict miracles associated with the saint.
The heart of modern Canterbury is still much as the hordes of medieval pilgrims would have found it, with narrow cobbled streets, quaint old buildings leaning outwards over the thoroughfares and venerable inns providing meat, bread and ale.
There's a further reminder of those times in The Canterbury Tales, an entertaining reconstruction of Chaucer's 14th century account of the stories told by a group of pilgrims, based appropriately in the old Church of St Margaret's.
Inside the cathedral, the stone floors still bear the marks of countless pilgrim knees. The glorious columns of the cathedral's nave still soar heavenwards, meeting in a majestic golden arch that fills pilgrims with awe. Among the magnificent tombs here are those of one of England's greatest warriors, Edward the Black Prince, who died in 1376 (his golden armour hanging overhead revealing him to be a shrimp) and Henry IV, who died in 1413 - the only king to be buried in the cathedral, apparently because he was a fan of St Thomas.
Nearby sits Augustine's Chair, official seat of the archbishops of Canterbury, a quiet reminder that this is an office which stretches back through 104 bishops.
But, sadly, the shrine which was once the focal point of the great cathedral is no longer there.
It was demolished in 1538 as part of the Reformation and Henry VIII - apparently carrying a grudge on behalf of his royal ancestor - ordered the remains of Becket to be cast to the winds so it could never be restored.
Today, only the candle burning in a roped-off area of stone paving marks the last resting place of his remains.
But it's hard to keep a good saint down and there are traces of Becket not far away in the Catholic Church of St Thomas of Canterbury.
There, above the altar in the small Martyrs' Chapel, are two reliquaries said to contain a fragment of Becket's bone, probably from a finger, and a small piece of his vestments.
An explanatory note suggests that these were originally presented to Cardinals from Rome, who attended the transfer of Becket's body from the crypt to the new chapels in 1220, and the reliquary with the bone was actually gifted in 1953 by Thomas Becquet, prior of the Abbey of Chevetogne and a collateral descent of the saint.
Canterbury contains many equally remarkable relics of its past, including Bronze Age tools, Neolithic barrows, Roman walls and a Norman castle. But it is best known as the cradle of Christianity in Britain - and hence New Zealand - due to the arrival in 596 of St Augustine (hence the name of the Archbishop's chair), sent by Pope Gregory the Great to convert the Saxons.
Remarkably, the place where Augustine first worshipped, St Martin's Church, is still in use, making it probably the oldest working church in Britain. Built in Roman times, or possibly by Saxons using Roman bricks, it was already being used as a chapel by Queen Bertha, the Christian wife of King Ethelbert of Kent, and so provided an ideal base for his fledgling church.
Later Augustine rebuilt another old church, Christ's Church, turning it into his cathedral, which over the centuries has grown into the magnificent structure which stands today. During recent renovations, the remains of the original cathedral were discovered below the floor of the nave.
Obviously a man of great energy, Augustine also founded an abbey, which once rivalled the great cathedral in size but was also destroyed during the Reformation. You can get a feel for its past glory by wandering round the ruins - but, unfortunately, much of the land is now used for two schools and closed to the public.
In the grounds of these venerable places of worship are the graves of many of the ancient kings of Kent, their bishops - including the original grave of Augustine himself - and their nobles.
But, ironically, perhaps the most visited tomb of all is that of Mary Tourtel, creator of Rupert the Bear, who was buried in the graveyard of St Martin's.
CHECKLIST
Getting there: Emirates has three flights a day from Auckland and one from Christchurch to Dubai, and flies from Dubai to several British airports including Gatwick in the south of England. Call 0508 364 728.
More information: General information on visiting Britain is at visitbritain.com.
Visitor information about Canterbury is at canterbury.co.uk and you can find out about the surrounding county of Kent at visitkent.co.uk.
Canterbury Cathedral's website is at canterbury-cathedral.org.
Jim Eagles visited Canterbury as guest of Visit Britain and Emirates.
