The Basic Rules are a place for you to start.
The Dungeon Master's D&D Basic Rules has a section on encounter building and modifying encounters (p.165-167) which will get you the same advice @Marius referenced in the DMG--at zero cost. Pay particular attention to "Party Size" on p. 166.
(The adventure contained in the Starter Set--The Lost Mine of Phandelver--does not contain notes on modifying it's encounters' levels.)
Note that the Basic Rules were revised in 2018. If you have the old Basic Rules, the referenced sections are on pages 56-58.
Next steps...
Take a look at Black Streams Solo Heroes, which has a lot of good advice for modifying adventures/campaigns/gm-style to suit the one-player experience.
Likewise, rpg.net has a good series of columns on Role-playing Duets that might give you some ideas.
But the best reference...
is to be up-front and honest with your players. Let them know you're not sure how things'll go and that you're very interested in their feedback. Some games are fun to play at the edge of survival, some players only want to play heroes who will embody the monomyth.
Most importantly:
All of the references above are distilled wisdom of other GMs over the years, wisdom borne of experience. It's all good, but none of it's going to be better than the experience you all get playing together and discussing what you like and what you hate. Use these as references for ideas, but use your players and yourself as your guide.
5E Realms lore does not specifically address this. 4E Realms lore, on the other hand, does. Their exact origin is subject to some debate, even among the elves.
Collectively, the elves are known as the Tel’Quessir6 ("The People"), a title that encompasses Eladrin, Wood Elves, High Elves, Sea Elves, and so on. It is 'widely accepted' that the Tel’Quessir are native to the Feywild1. It is believed that they first immigrated to Abeir-Toril over a hundred millenia ago, while the exact date is uncertain it is believed to have been prior to −24,000 DR, with some manuscripts indicating it was as early as −30,000 DR, the same time in which dragons were setting up their empires (Note: Source on this is from AD&D)4. The Wild Elves were the first to arrive.1
The exact origin of the Elvish race is subject to some debate. Some ancient manuscripts suggest that they were the result of a battle between Gruumsh and Correlon that occurred in the Feywild. During said battle, Correlon was injured and his shed blood became the Eladrin.2. Other types of elf, not liking the superiority this implies in that the Eladrin are the first and 'purest' of elves, disagree with this theory3. Another theory postulated is that since the Feywild tends to be a 'reflection' of the Prime Material Plane, and thus produces creatures that are echoes of creatures from the Prime, it is also possible that the Elves are one such echo... though what, exactly, they would be an echo of is uncertain1.
Perhaps also worthy of mentioning is the current in-lore explanation for how the Feywild (also known as Faerie) is treated according to current lore. The Feywild was created as it exists today, a 'lighter' echo of the Prime Material Plane. It was made by Primordials in the far distant past, before recorded history begins.5
For a while, the Feywild existed alongside the Prime and interaction between the two was common. But, at some point in ancient history, the two drifted apart and travel between them became increasingly difficult2. This is the explanation for why, in prior editions of D&D, the Feywild didn't "exist" in the Forgotten Realms. It was out there, you just couldn't get to it.
The Spellplague (the event that 'kicks off' 4E in the Realms) changed that, reshuffling the cosmology to drag the Feywild back into close proximity, and placing it in the accessible location it resides in today5.
Checking through published 5E Realms material, it does not contradict any of this... the 5E Realms cosmology is essentially the same as the 4E Realms cosmology, and the lore established within 4E is considered to hold true in 5E, except where directly overridden. And even then, the changes are usually justified in-lore (such as the array of gods and goddesses getting rejiggered due to Ao meddling with them).
The primary lore sources we have in 5E Realms are the Sword Coast Adventurer's Guide, the DMG, and published adventures. None of these go into detail on the origins of the Elves, or even on the specific nature of the Feywild... thus we may assume the lore of older sources hold true.
Sources
1 Brian R. James and Ed Greenwood (September, 2007). The Grand History of the Realms. (Wizards of the Coast)
2Bruce R. Cordell, Ed Greenwood, Chris Sims (August 2008). Forgotten Realms Campaign Guide. (Wizards of the Coast)
3James Wyatt (December 2007). Dragon 361: A Fractured Family. Wizards of the Coast.
4Steven E. Schend and Kevin Melka (1998). Cormanthyr: Empire of the Elves. (TSR, Inc) NOTE: AD&D source
5Dungeon Master's Guide 4th edition. (Wizards of the Coast)
6Kim Mohan ed. (2015). Sword Coast Adventurer's Guide. (Wizards of the Coast)
Best Answer
Minsc is mentioned, although doesn't feature as an actual character, in Baldur's Gate: Descent into Avernus
As you noted, Minsc is mentioned in passing in the D&D Encounters module Murder in Baldur's Gate (written for 3.5e, 4e, and the "D&D Next" 5e playtest), specifically that there is a statue of him and Boo in the The Wide, the great marketplace in the Upper City. This detail is featured again in BG:DiA, where it is explained that what had been presumed to be a statue was, in fact, actually the petrified characters themselves - and at some point in the years between MiBG and BG:DiA they were cured and set to wandering the world again:
However, he does not otherwise feature in the adventure. It seems likely that this specific detail is a way of keeping the character available for use in future 5e material - establishing that Minsc & Boo have been petrified for the last hundred or so years neatly avoids the issue that a normal human like Minsc who was active in the late 1300s DR should have been dead (or at best extremely old) by the 1480s and 90s, when 5e's FR material is set. (As a miniature giant space hamster, we can probably assume that Boo's lifespan isn't an issue.)
He's also seen as an easter egg in an illustration in Waterdeep: Dragon Heist
The section of W:DH's introduction describing The Yawning Portal tavern includes a cartoon illustration of the tavern's layout full of depictions of famous characters of D&D and Realms lore enjoying the tavern's attractions. Minsc and Boo are number 22, located between Volothamp "Volo" Geddarm and Krydle (a character who features alongside Minsc & Boo in the IDW D&D comics).
Of course, this depiction doesn't imply anything canonical about Minsc & Boo's activities or whereabouts.
Minsc and Boo feature in WotC's The Descent livestream event
If you widen the net a little, Minsc doesn't feature in any other officially published written material, but his character did appear in Wizards of the Coast's 2019 live play livestream event The Descent, in which he is found trapped in Avernus with a motley collection of other heroes and eventually manages to escape back to Baldur's Gate (though the canonicity of this adventure may be questionable).
Here he was played by Jim Zub, who had previously written the character in the IDW D&D comics. Boo was played by Matt Mercer, of Critical Role.