Byron has several functions in Arcadia, both as the historical poet and as an individual who exercises an influence on events in the plot.
Firstly, Byron was a real person. The references to his presence at Sidley Park encourage the audience to link with reality the events depicted there.
The unseen but nonetheless present Byron interacts with characters whom we do see in the 19th century scenes of the play. Initially he is unidentified, but the nameless 'friend' who shoots a pigeon during scene one is subsequently confirmed as Byron in scene three. Lady Croom is in love with him (or at any rate, fancies him), which disappoints Septimus, who is in love with her. Lady Croom laments the poet's intention to leave England—we see that this intention exists well before any duel is fought, and it seems that Byron is simply looking for adventure in foreign parts.
Although Bernard's hypothesis that Byron fights the duel is incorrect, Byron does nudge Chater into renewing hostilities with Septimus. Thomasina reports that Byron gave Septimus away as the author of the scathing review of Chater's poem. Furthermore, Byron apparently requested a free copy of Chater's new book in order to include the unfortunate Chater in the next edition of Byron's 'English Bards and Scotch Reviewers', which will of course mean further ridicule.
In scene six, we learn that Byron has departed from Sidley Park taking Septimus' book with him. Byron did, indeed, indulge in a liaison with the equally invisible Mrs Chater, which has annoyed Lady Croom into packing them both off the premises and incidentally ensuring that no duel actually takes place. There is even a letter left by the departing Byron for his friend Septimus, which might be supposed to have contained enough information to scupper Bernard Nightingale's beloved theory forever. Septimus burns it along with his own explanatory letters. Later still, it seems that Thomasina has a crush on Lord ByronÉ who has been seen posing at the Royal Academy for a picture which Hannah will use as the cover for her book.
So much for Byron as a participant in the plot. As a historical figure he is also of considerable significance to Bernard and Hannah. To Hannah he is a necessary part of her research into Lady Caroline Lamb; to Bernard, Byron is the means by which he hopes to achieve academic fame—and also, in the light of the actual events which the audience sees—a most unfortunate red herring!
The first introduction of Byron's name is a surprise to the audience, as the character has not appeared in 19th-century Sidley Park (or in the cast list). From what the audience already knows, it is clear that Bernard's theory is wildly wrong. However, we then discover that Septimus' book was found in Byron's possession, and the scene does fix the two men as close contemporaries, if not necessarily as friends.
Subsequently, Bernard's theories about Byron are shown to be somewhat closer to the truth than first appeared to be the case. He assumes a seduction of Chater's wife, leading to a duel. We already know that Mrs Chater was seduced, in fact by Septimus, and that there is to be a duel. And, to confirm the discovery of Byron's presence in scene three, Valentine produces the game book with proof that Bernard can use. (Proof which is, arguably, based on a lie, since Augustus later insists that he and not Lord Byron actually shot the hare!)
In scene five, quotes from Byron's own letters are used within Bernard's lecture. Bernard is in high gig, despite Hannah pointing out that Byron was talking of going abroad well before he visited Sidley Park. Eventually, Bernard's well-publicised theory is dashed by a discovery by the more painstaking Hannah, leading to humiliation for the one and undoubted satisfaction for the other.
Byron's name is linked throughout to the confusion of facts, to the way evidence is lost (quite literally turned to ash, in the case of his letter to Septimus). Through Bernard's misapprehensions about the past, we see the impossibility of 'winding the film backwards'. And in scene seven, Hannah's own quotation from one of Byron's poems connects the poet to the theme of entropy—like Thomasina, the genius of the poet has made an intuitive leap to understanding the way the universe works.
It has also been speculated that Byron's own daughter, the mathematically gifted Ada, may have been the inspiration for the character of Thomasina. In fact, Stoppard has explicitly denied having a model for this character, saying that he thought it would be interesting and unusual in 1809 to have a young girl being tutored in mathematics, but that in other respects, Thomasina is an ordinary young woman, which is what makes her appealing.