1491 DR
Ed Greenwood gave an answer to this on Twitter:
Lost Mine is officially a bit nebulous in date for DM convenience, though there are detailed internal WotC timelines so as to keep things straight (i.e. novels). Safe to say: 1490s DR.
Of course, his tweets aren't official source material, but they give a small glimpse behind the curtain at designer intent, and are a reminder that all these dates have an important context: the metaplot.
The Forgotten Realms metaplot
The Forgotten Realms setting has a metaplot that continues to advance chronologically with new source material, and new editions of the game. For example, when the 4th edition of D&D came out, the metaplot advanced by about a century.
The first 4th Edition campaign setting releases later this summer, with a new look at the Forgotten Realms. If you've read the previews, you'll know by now that major changes have affected the Realms: starting with the Spellplague and the 100-year advancement of the timeline.
4e material
The century of advancement in the setting resulted in a lot of drastic changes to the world. The 4e Forgotten Realms Campaign Guide (2008) notes:
Ten Important Facts
The following points describe the biggest changes
to the world of Toril since the previous edition of
the FORGOTTEN REALMS Campaign Setting. If you are
familiar with the setting, these will summarize the
major events in the world since 1374 DR, the Year of
Lightning Storms. If you are new to the setting, this
information will give you the basic background that
most inhabitants know.
1. Roughly a hundred years have passed in the world
since the previous edition of the campaign setting. The
current year is 1479 DR, the Year of the Ageless One.
2. The Spellplague has drastically altered the cosmos.
The Spellplague broke out in 1385 DR (the Year of
Blue Fire), the result of unfettered wild magic on the
death of the goddess Mystra. Whole countries are
gone, especially in regions south of the Sea of Fallen
Stars. [...]
And so on. Flying islands in the sky called "earthmotes." A continent that wasn't there before on the other side of the Trackless Sea. Like New Coke, all these 4e changes were rapidly done away with in the transition to 5e, as a result of a cross-media event called "The Sundering" announced in 2012.
In 2013, the D&D Encounters adventures began to take part in The Sundering plotline starting with D&D Encounters season 15: Murder in Baldur's Gate. Being in a transitional period, the module was written to be playable with any of three editions of D&D: 3.5e, 4e, or 5e (then called "D&D Next").
5e material
"A Brief History" in Sword Coast Adventurer's Guide directly alludes to the events in Murder in Baldur's Gate and The Sundering plotline (calling it "The Second Sundering" since there was also an earlier cataclysm by that name):
The first indication of new turmoil came in 1482 DR, when Bhaal, the long-dead god of murder, was reborn in Baldur’s Gate amid chaos and bloodshed, leaving two of the city’s dukes and many of its citizens dead.
And it caps off the history of the Realms with establishing a present day status quo that has hit the reset button on most of those setting alterations from 4e, with everything noticeably "much like it was" before "that terrible time." The present day being 1489 DR or later:
Early in 1487, earthquakes and volcanic eruptions abounded for months, as if the whole world was convulsing. Rumors spread of chasms caused by the Spellplague suddenly vanishing, and stories circulated of known destinations being farther away from one another, as if the world had quietly added miles of wilderness to the distance between them. Word began to spread of places and peoples not heard from since the Spellplague. It became apparent that some of the effects of that terrible time had been reversed.
By 1489, many of the wars that began during the Sundering had ground to a close. Other conflicts arose, and mighty threats still imperiled the world, but the deities ceased interfering with the world through their Chosen. The gods were no longer silent but quiet, and in many places new priesthoods arose to interpret the gods’ now subtle signs.
The world today seems a place filled with new lands and opportunities, where those who dare can leave their mark. Students of history and those elves and dwarves who recall the past that short-lived humans see as distant perceive a world much like it was over a century ago.
So this is the context in which Ed Greenwood said that LMoP definitely occurs in the 1490s: that's roughly where the metaplot left off when 5e began.
So when is Lost Mine of Phandelver?
Official material wise, Acquisitions Incorporated contains an adventure explicitly set five years after Lost Mine of Phandelver:
HAVE WE BEEN HERE BEFORE?
The town of Phandalin will be recognizable to any players familiar with the Lost Mine of Phandelver adventure. This version of Phandalin is set five years or so after that adventure, making use of many of the original Phandalin locations and NPCs even as it adds new places and new characters to the mix. If you have your own version of Phandalin in your campaign, make any adjustments necessary to this episode for continuity.
The same adventure is also set five years after Princes of the Apocalypse:
The Princes of the Apocalypse adventure contains more information on Red Larch and the surrounding region. As with Phandalin, this version of Red Larch is set about five years later than that earlier adventure.
And Princes of the Apocalypse is explicitly set in 1491 DR:
TIME IN THE REALMS
The people of the Forgotten Realms number the years by the Dalereckoning calendar, which is abbreviated “DR.” Years are also given names, which are drawn from the writings of a great seer from long ago. The adventure is set in 1491 DR, the Year of the Scarlet Witch.
Each month consists of three ten-day long weeks called tendays. People refer to tendays the way people in other worlds refer to weeks.
Since Acquisitions Inc establishes that LMoP and PotA are set around the same time, LMoP occurs in 1491 DR.
Given that as a general rule, newly published material seems to be set in the 'present day' of the ever-advancing Forgotten Realms metaplot, 1491 DR is a lot more credible than the seemingly contradictory line in LMoP that alludes to the eruption of Mount Hotenow in 1451 occurring only 30 years prior. The year 1481 DR was the 4e era, before The Sundering and the transition to 5e had even begun.
