Within the published book
There really isn't anything here to go on. The arrow gets mentioned in this scene, gets very briefly mentioned by Gandalf in the next chapter, and then appears again in the chapter after that when the Rohan scouts find Hirgon's corpse.
‘Yes, he will come,’ said Gandalf, ‘even if he comes too late. But think! At best the Red Arrow cannot have reached him more than two days ago, and the miles are long from Edoras.’
The Lord of the Rings - Book V Chapter 4 - "The Siege of Gondor"
‘This, lord: they were errand-riders of Gondor; Hirgon was one maybe. At least his hand still clasped the Red Arrow, but his head was hewn off.
The Lord of the Rings - Book V Chapter 5 - "The Ride of the Rohirrim"
None of this really adds any additional information, just that Gandalf is also familiar with it.
In other works
The red arrow gets a very brief mention in a footnote to "The Battles of the Fords of Isen", but this is a reference to when Théoden was sent the arrow in the book, and as such also doesn't add any additional info.
When Théodred was at last slain Saruman's commander (no doubt under orders) seemed satisfied for the time being, and Saruman made the mistake, fatal as it proved, of not immediately throwing in more forces and proceeding at once to a massive invasion of Westfold ... The Ents and Huorns, with the aid of such Riders of the East-mark as had not yet been engaged, might have destroyed the forces of Saruman in Rohan, but the Mark would have been in ruins, and leaderless. Even if the Red Arrow had found any one with authority to receive it, the call from Gondor would not have been heeded - or at most a few companies of weary men would have reached Minas Tirith, too late except to perish with it.
Unfinished Tales - "The Battles of the Fords of Isen"
From the drafts in The History of Middle-earth
The only information available from The History of Middle-earth is that the red arrow was introduced in the first full narrative Tolkien ever wrote describing the exchange with the Théoden and the messenger(s), though it was not mentioned in any of the eight or so (very brief) outlines that had previously mentioned there being a messenger. Christopher does not actually present any of the versions of the exchange but the only difference about the arrow that he mentions is that the first draft had a green-feathered arrow instead of a black-feathered arrow. (See The War of the Ring, Part 3, Chapters 2 and 5, especially footnote 7 to chapter 5)
Tolkien's inspirations
Hammond and Scull suggest that Tolkien may have here been inspired by Morris.
798 (III: 72). In his hand he bore a single arrow, black-feathered and barbed with steel, but the point was painted red. - In A Tale of the House of the Wolfings by William Morris the Wolfings are summoned to war against the Romans in part by a messenger who carries ‘the token of the war-arrow ragged and burnt and bloody’ (Chapter 2).
The Lord of the Rings: A Reader's Companion - Book V Chapter 3 - "The Muster of Rohan"
Some random person's masters thesis suggests that Tolkien was inspired by Cynewulf
Denethor, steward of Gondor, calls the Rohirrim to battle by sending a rider
bearing a red arrow as a “token of war” (Return 64), a parallel with Cynewulf’s Elene in which Constantine summons “his heroes to war against the foes” through “dispatch of the arrow” (43-45).
Recreating Beowulf’s “Pregnant Moment of Poise”: Pagan Doom and Christian Eucatastrophe Made Incarnate in the Dark Age Setting of The Lord of the Rings