You have to be careful when talking about the lives of the dead, because they can't do the edits. Although your question is asking about historical accuracy, I want to address it from another perspective. Personally, I have an undergraduate degree in Anthropology while I am currently finishing my master's in History. With that, I see questions like this pretty often, and there always the same pitfall. How do we categorize, classify, and label the relationships of historical figures in cultures removed from our own?

Both Hume and Smith lived in Scotland in the 18th Century. Hume dying in 1776 and Smith dying later in 1790. Both of them were well educated (Smith educated at Oxford and Glasgow and Hume at Edinburgh though he did not graduate) and obviously, accomplished in their academic writings. I bring this up, because it highlights our first problems, what can we say about relationships of people so far removed from us?

We know that Hume and Smith were friendly and were even involved in the same clubs, in fact, a book titled “The Infidel and the Professor” published by Princeton University Press and written by Dennis Rasmussen a professor out of Syracuse University, talks about the friendship of the pair. Rasumussen mentions, that even though they lived in separate cities, they both still engaged with each other in this interview.

So we know they were close and we know they were friends, but what about your question, is the theory of them being lovers credible and when did it first appear?

To answer the second question first, I have no idea. I cannot really find a date for when the idea came about and perhaps other commenters can find better information than I can.

As for the question of creditability to the theory of them being lovers, I have more to say. Firstly, in trying to classify Hume and Smith as being lovers we are trying to apply our modern label to the lives of two men who lived over 240 years ago. This is always one of the issues with biography or even microhistory, in what Jill Lepore in her article “Historians Who Love Too Much” says when writing about people we forget their society and when writing about society we forget about the people. I take this to mean, that we have to remember both when writing about either, because society and people influence each other. The theory of them being lovers now has to cross time, culture, and class status to even try to find an answer. The problem is that modern ideas on romance, love, and relationships has changed so vastly (in just the past 40 years) and what is viewed as “acceptable” changes so dramatically between cultures that it is impossible to fully understand the exact nature of their relationship. That’s our major problem here… we are asking if Hume and Smith were lovers, but did Hume or Smith see themselves as lovers?

This last point is spoken to in Robert Mills article “Queer is here? Lesbian, Gay, Bisexual and Transgender Histories and Public Culture” in History Workshop Journal (autumn, 2006) where he talks about the complicated and changing identities of men who have sex with other men not considering themselves as gay and where they would fall in a history of LGBT people. Should they be included in gay histories despite them excluding themselves? This is the complexity and issue with such assertions. Our understanding of LGBT identity is modern while our knowledge of Hume and Smith is over 240 years old.

However, this all really comes to one main point: without direct unambiguous evidence, we just cannot know, to state anything otherwise is conjecture. As you mentioned a lot of their personal corresponsive was destroyed (this is supported in “The Death of Hume” published 1995 in The Wilson Quaterly by Stephen Miller), eliminating a lot of data we could use to try and understand their relationship better and to understand what they saw their relationship to be. In all, it is the responsibility of the theory to prove their argument and without more complete data we just cannot know. It should be said though, that this ends up artificially decreasing representation of LGBT people, erasing their stories and making it seem that its more of a modern phenomenon, when we know that not to be true. It’s challenging, but to go back and apply modern labels to historical figures would cause greater harm to representation in modern historical literature as well as bring the discipline of history under fire for revisionism.

Edit: I changed the language of my first sentence after some very insightful comments and realizing that it came across as negative and attacking to the LGBT community, something that I had no intention of doing.