What is the basis for the belief that the Apostle John died around AD 100?

The early testimony of Irenaeus

Irenaeus of Lyons explicitly tells us that John was still living at the time of Emperor Trajan's accession (AD 98):

[T]he Church in Ephesus, founded by Paul, and having John remaining among them permanently until the times of Trajan, is a true witness of the tradition of the apostles. (Against Heresies 3.3.4)

And he [John] remained among them up to the times of Trajan (Against Heresies 2.22.5)

Irenaeus was a pupil of Polycarp who was a disciple of John. Furthermore, Irenaeus grew up in early 2nd century Asia Minor, in a world saturated with John’s influence. He also studied the works of Papias (another disciple of John). Irenaeus is in an extraordinarily good position to know what he's talking about here.

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Other writers

Ignatius. Ignatius gives us a useful upper bound. Writing in approx. AD 107 he addressed the Ephesians and made no reference to John (his mentor!), even though he did make specific reference to other less-prominent leaders in his epistles. This is inconceivable if John is still there.

Clement of Alexandria speaks of John's return from Exile after the death of the tyrant who banished him (the tyrant is unnamed but has often been understood to be Domitian (died AD 96) based on the later writings of Eusebius and others (see What Rich Man can be Saved?).

Eusebius puts John in Ephesus during the reign of Nerva (AD 96-98)

10. But after Domitian had reigned fifteen years, and Nerva had succeeded to the empire, the Roman Senate, according to the writers that record the history of those days, voted that Domitian's honors should be cancelled, and that those who had been unjustly banished should return to their homes and have their property restored to them.

11. It was at this time that the apostle John returned from his banishment in the island and took up his abode at Ephesus, according to an ancient Christian tradition. (HE 3.20.10-11)

Jerome puts John's death during Trajan's reign (AD 98-117):

But Domitian having been put to death and his acts, on account of his excessive cruelty, having been annulled by the senate, he [John] returned to Ephesus under Pertinax and continuing there until the time of the Emperor Trajan, founded and built churches throughout all Asia, and, worn out by old age, died in the sixty-eighth year after our Lord's passion and was buried near the same city. (De Viris Illustribus ch. ix)

The 68th year after the Lord's passion is insufficiently precise to determine an exact year (inclusive vs. exclusive counting, crucifixion in 30 vs. 33 vs. another year), but it does get us to the late 90's or early 100's.

Papias. Some have attributed to Papias the idea that John died a martyr's death around the same time as his brother James (died ~AD 42-44). This argument is untenable. Although the writings of Papias have been lost, Irenaeus had read Papias and Irenaeus clearly knew that John did not die in the 40s. Furthermore, Papias knew John personally (see Irenaeus Against Heresies 5.33.4), and Papias wasn't born until years after the death of James.

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History or tradition

The OP made reference to tradition, and I believe it would be helpful to share an observation I've made many times in the study of Greco-Roman history. When a well-documented non-Christian historian recorded something from this period, it's usually called history. When a well-documented Christian historian recorded something from this period, it's usually called tradition. This inconsistent nomenclature in academia is both misleading and unhelpful.

Men like Papias, Irenaeus, Clement of Alexandria, and Eusebius were theologians yes, but historians as well. Some are uncomfortable granting them "historian" status because they make supernatural claims. Here again an inconsistency among modern historians is laid bare: with the exception of Thucydides all of the major Greco-Roman historians make supernatural claims.

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Death or translation

Over the years many have held the view that John never died. My own faith teaches this view. This is one of the common interpretations of John 21:22-23.

It is noteworthy that our most important source, Irenaeus, in fact says nothing about John's death, but merely speaks of him staying with the saints of Ephesus until a certain time.

It is possible to read everything in this post and assume that John died around AD 100; it is also possible to read everything in this post and assume that John's public ministry closed around AD 100 but that he did not die.

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Conclusion

We do not need to appeal to symbolic interpretation of Revelation to adduce the year AD 98; Irenaeus gives us the information explicitly. There are indeed many who have interpreted the kings/rulers/kingdoms of Revelation to indicate that Domitian was the present-day ruler when Revelation was written...but others have had no difficulty interpreting Revelation to say that Nero or Vespasian was emperor at the time, not Domitian.

Neither need we appeal to anonymous tradition--we have a very well-placed source in Irenaeus, who indicates John was still living in AD 98. This statement by Irenaeus is corroborated repeatedly by other historians.

We are not given the year 100 explicitly; it is a useful number to round to. John's ministry ended between AD 98 (Trajan's accession) and the writing of the epistles of Ignatius (~AD 107).