Foreword: I am using 3 characters from Forgotten Realms, just to illustrate the scenario. The question is not about FR, but about skill mechanics and DC. Also history is used below just as an example.
Given 3 characters with the same history skill total (same ability modifier, same proficiency), each of them alone (no assist) facing the same history skill check:
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The 1st one is from a radically different background, and shouldn't have ever even heard about the subject in his homeland.
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The 2nd one is from a neighboring background, that is indifferent to the subject.
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The 3rd character's background has intricate intimacy with the subject, and could have had several chances to learn about the subject during his early days.
The subject are the ruins of an ancient elven kingdom far in the north-west: The elven realm of Eaerlann.
Lets give a face to our historians:
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A dwarf librarian from a dwarf realm underneath Var the Golden in the far south-east. He has never left his lands until now, and had little contact with elven history.
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A human mage from Waterdeep. He traveled a bit around the region, it would not be a surprise if he learned some lore about Eaerlann.
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An elven archeologist from Evereska. His homeland is right next to where the fallen kingdom was, some survivors from that place ended up in his homeland, and has all the interest in elven lore.
Given those 3 characters, how should the DM adjucate the familiarity of each character on the skill roll:
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Do not change anything. Everyone rolls the same DC as a standard skill check.
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Grant advantage/disadvantage based on familiarity. The dwarf has disadvantage, the human rolls normal, and the elf has advantage.
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Modify the skill DC. It is a hard roll for the dwarf, a moderate for the human and an easy one for the elf.
Of course you are not bound by these 3 choices above, but please do state the reasons.
(I also suspect that RAW has nothing on the matter)
Best Answer
There is no right answer to this question, but there is a guideline:
I don't have the DMG/PHB right now to point you the exact quote but in short: The DM decides what to roll.
There are conditions that might favor what you're trying to do (intimidating thugs after you just rolled a critical and killed their boss), and there are conditions that makes things harder (trying to talk your way out of a murder acusation after being found with the murderer's weapon).
Note that none of this situations are defined in the rules, it all comes down to the DM's adjudicating the situation.
There are rules for combat that tells you what to roll and when to roll, there are guidelines to challenges but ultimately, it's all up to the DM, so everything I say after this is utter bullshit if your DM says otherwise but;
Option 2 sounds quite about right, but, if I were in your DM's shoes, I would just give the elf the answer right away. Yes, you DM might rule that this particular task requires no roll due to the conditions.
It makes complete sense that he should know this alredy, hell, in fact it hardly makes sense that he doesn't, if a player invested on some sor of background, he should have that reflected on his knowledge about the world and how he interacts with it.