Review of the main sources
Let us take a look at the earliest sources which give an account of this story involving Heracles, Hesione, her father Laomedon, and her brother Priam. It is first briefly referenced to in Homer's Iliad:
Of other sort, men say, was mighty Heracles, my father, staunch in fight, the lionhearted, who on a time came hither by reason of the mares of Laomedon with but six ships and a scantier host, yet sacked the city of Ilios and made waste her streets.
Il. V.638-642 Transl. A.T. Murray
The first lengthy account of the story is given by Diodorus Siculus in his Library (I century BC). Here I am quoting only the passage which is relevant to the question:
For when Heracles was on the expedition with Jason to get the Golden Fleece and had slain the sea-monster, Laomedon had withheld from him the mares which he had agreed to give him [...] Laomedon then withdrew and joining combat with the troops of Heracles near the city he was slain himself and most of the soldiers with him. Heracles then took the city by storm and after slaughtering many of its inhabitants in the action he gave the kingdom of the Iliadae to Priam because of his sense of justice; for Priam was the only one of the sons of Laomedon who had opposed his father and had counselled him to give the mares back to Heracles, as he had promised to do. And Heracles crowned Telamon with the meed of valour by bestowing upon him Hesione the daughter of Laomedon, for in the siege he had been the first to force his way into the city, while Heracles was assaulting the strongest section of the wall of the acropolis.
Diod. Sic. IV.32 Transl. C.H. Oldfather
Note that here there is no mention of the veil: no ransom is payed for Priam's freedom. Instead, in the Fabulae by Hyginus (I century BC) we have an indication that a ransom was payed for Priam's freedom, but again no mention of Hesione's veil:
When many girls had been devoured, and the lot fell on Hesione, and she was bound to the rocks, Hercules and Telamon came there, the Argonauts being on their way to Colchis, and killed the monster. They delivered Hesione to her father on condition that when they returned they should take her with them to their country, as well as the horses which walk over water standing ears of grain. Laomedon defaulted in this, too, and refused to give up Hesione. And so Hercules, assembling ships to attack Troy, came and slew Laomedon, and gave the kingdom to his infant son Podarces, who was afterward called ἀπὸ τοῦ πρίασθαι [from being redeemed], Priam. He recovered Hesione and gave her in marriage to Telamon. Their child was Teucer.
Hyg. Fab. 89 Transl. M. Grant
The first surviving mention of the veil appears in Lycophron's Alexandra (II century BC) in which Priam is described as "a poor prisoner ransomed for his sister's veil". This is corroborated by the Bibliotheca of Apollodorus (I-II century CE):
The siege once laid, Telamon was the first to breach the wall and enter the city, and after him Hercules. But when he saw that Telamon had entered it first, he drew his sword and rushed at him, loath that anybody should be reputed a better man than himself. Perceiving that, Telamon collected stones that lay to hand, and when Hercules asked him what he did, he said he was building an altar to Hercules the Glorious Victor. Hercules thanked him, and when he had taken the city and shot down Laomedon and his sons, except Podarces, he assigned Laomedon's daughter Hesione as a prize to Telamon and allowed her to take with her whomsoever of the captives she would. When she chose her brother Podarces, Hercules said that he must first be a slave and then be ransomed by her. So when he was being sold she took the veil from her head and gave it as a ransom; hence Podarces was called Priam.
Apollod. II.6.4 Transl. J.G. Frazer
Much later, Tzetzes adds the detail that the veil was golden in his Ad Lycophronem (XII century CE):
when Troy was destroyed by Hercules, Priam, then called Podarces, was led captive; but his sister Hesione, giving her golden veil, bought him, hence he was called Priam and left him in his homeland hoping that it would be restored by him, which also happened. She was given to Telamon, from whom Teucer was born.
Ad Lyc. 337 Ed. E. Scheer
Conclusions
Apart from Tzetzes' late account, there is no indication that the veil was particularly valuable, and there is certainly no evidence that the veil had anything to do with Aphrodite.
However, it should be noted that in Greek epic a woman's veil was traditionally associated with marriage and chastity, and in all the accounts of the story Hesione is given in marriage to Telamon by Heracles. One might then conclude that it isn't just the literal veil that Hesione is giving as ransom for Priam, but also herself.
On the symbology connecting veils, chastity and marriage, I recommend checking out David West's article Shipwrecked Spouses: Leukothea’s Veil and Marital Reunion in
The Odyssey and references therein. There, incidentally, West cites a passage from the Iliad involving a Trojan woman, a veil, and Aphrodite: it's the moment in which Andromache, having discovered Hector's death, throws away the veil that Aphrodite had gifted her on the day of her wedding. This, according to West, symbolizes the end of her marriage, and the violation that awaits her.
