Let's bear in mind that according to the sources referenced in the Wikipedia page Manimekalai, this book was written either before the 5th century CE, in the 6th century CE or latest 950 CE according to different experts.
Anne E. Monius is a Professor of South Asian Religions at Harvard University and "a historian of religion specializing in the religious traditions of India. Her research interests lie in examining the practices and products of literary culture to reconstruct the history of religions in South Asia." She wrote the book "Imagining a Place for Buddhism: Literary Culture and Religious Community in Tamil-Speaking South India".
She talks about the book Manimekalai and 1616 prediction in this book, for which you can find the excerpt here. Please pay attention to the last paragraph on page 94, and the next two pages.
Her best guess is that since the Buddhist calendar usually starts counting from the final passing (parinirvana) of the Buddha (also see this answer), this refers to the year 1616 on that calendar. She is not sure about how this number was derived.
From the Wikipedia page on the Buddha, the date of the passing away of the Buddha is given as either 483 BCE or 400 BCE. That puts 1616 at 1133 CE or 1216 CE. This is definitely centuries after the authorship of the Manimekalai. So, that's quite sensible. Prof. Monius stated that this is more specific and substantially earlier than the estimates given in the Pali sources, for the coming of the next Buddha.
According to accesstoinsight.org's page on the Chronology of Theravada Buddhism:
Year 1 of the Buddhist Era calendar is the year of the Buddha's
Parinibbana (death and final release), which occurred in the Buddha's
eightieth year (480 BCE according to the "historical" timeline; 544
BCE by tradition).
If we use 544 BCE as the year of the Buddha's parinirvana, then 1616 would be 1072 CE, which is still centuries after the authorship of the Manimekalai (if we assume that it was authored in the 6th century CE).
Also interesting is this excerpt from the book by Monius (on pages 95 - 96):
In fact, if one looks beyond, or more appropriately before, the Pali
literature that obviously concerns itself with the future Buddha,
scattered bits of evidence can be found to suggest that the
Manimekalai was not alone in the sixth century in imagining the
arrival of a living Buddha within centuries rather than billions of
years. In the various renditions of the life of Buddhaghosa, for
example, it is reported that the monks of the Mahavihara greeted the
composition of the Visuddhimagga by crying out to its author, "Without
doubt he is Metteyya!" (nissamsayam sa metteyo). Although such a
statement might easily be interpreted, on the one hand, as hyperbolic
praise, likening only rhetorically the genius of Buddhaghosa to
the wisdom of the future Buddha, on the other hand, such an
identification might well depend on the expectation, like that of the
ascetics confused by the miraculous events accompanying Aputtiran's
rebirth in the Manimekalai (xv.23-35), that Metteyya's arrival on earth
can happen at any moment. Such scattered phrases are certainly
difficult to evaluate. Much more interesting and potentially useful
for the consideration of the Manimekalai's vision of the future, given
the central place of the wondrous almsbowl, Amutacurapi, in the text,
is a story told by the fifth-century Chinese pilgrim to India and Sri
Lanka, Faxian (Fa-hsien), concerning the begging bowl of the Buddha
and the coming of Metteyya/Maitreya to earth.
You can read on in the same section, the story of Faxian, and how he obtained the story of the Buddha's almsbowl from an Indian monk.
Monius writes further on pages 96 - 97:
Faxian's story of the Buddha's almsbowl and its intimate connection to
the coming of Maitreya provides a potentially enlightening parallel
to the narrative of the bowl and the coming of the Buddha found in the
Manimekalai. Such a parallel, especially given the relative proximity
to the time and place of the Manimekalai's composition, of Faxian and
the Indian monk who tells the story at the Sri Lankan Mahavihara, may
help to illumine the Tamil text's vision of the future and the
community to form around the coming Buddha.
Based on this, I would guess that the Manimekalai is a literary work of fiction, based on Faxian's story that came not long before the authorship of the Manimekalai. While it is unknown how Manimekalai's author derived the number "1616", we can speculate that he intended his readers to look forward to the coming of the future Buddha within a few centuries, perhaps to motivate the Tamil-speaking Buddhist community at the time, and also to kindle the interest of the non-Buddhist Tamil-speaking people, in Buddhism.