TL;DR: Most likely no.

Before you read this whole post, you should be aware that any warrior's connection with Hestia is extremely unlikely given that Hestia the Goddess of the Hearth. She was originally an Olympian, but gave up her Olympic seat to Dionysus. I've read various suggestions about this, but one of the biggest limitations Hestia had over the other gods and goddesses is that she couldn't go on any adventures. Indeed, by her very nature, she is confined to the home. There is no hearth in war, except that one to which you long to return. There is no Hestia on the battlefield; she does not come to you in dreams.

However, Vesta is described by Ovid as the Earth itself and as having a perennial fire. Vesta is etymologically also connected to the Vedic traditions of fire and burning, so perhaps that alone is a connection enough for you.

That said, I'm going to assume that Psalm 102:3 doesn't hold your answer?

For my days are consumed like smoke, and my bones are burned as an hearth. (KJV)

I will also assume that this isn't a reference to the Mesha Stele, which has a suggested reading that refers to the "altar hearth of David."

A last assumption, the 4 Vesta asteroid is not somehow related? I don't know of any David and Goliath asteroids, but perhaps that's relevant to the setting/ plot/ weapon? Well, with these assumptions, we can chase the more interesting rabbit of mythological parallels!

It is undisputed that the Goliath story is an interpolation. This is obvious by being inconsistent with the later mentioned death of Goliath by Elhanan, though there are other reasons.

I might also mention that Goliath wasn't a giant of 9', but closer to 6'6". According to the LXX and the DSS (4QSamb), Goliath stands 4 cubits. It's also well known that skeletons from the time period put the general male population no taller than 5'5", so Goliath could still be quite the "giant" by comparison.

So the first obvious question is, "Does David and Goliath have a Greek parallel?" Finkelstein and Galling both say that Goliath wore Greek armor, but /u/Flubb has an excellent dissection of the arguments on /r/AcademicBiblical. There is, however, a case to be made in favor of a parallel with Nestor and Ereuthalion. Here's the 1924 English translation by A.T. Murray (Iliad 7:132-160):

I would, O father Zeus and Athene and Apollo, that I were young as when beside swift-flowing Celadon the Pylians and Arcadians that rage with spears gathered together and fought [135] beneath the walls of Pheia about the streams of Iardanus. On their side stood forth Ereuthalion as champion, a godlike man, bearing upon his shoulders the armour of king Areithous, goodly Areithous that men and fair-girdled women were wont to call the mace-man, [140] for that he fought not with bow or long spear, but with a mace of iron brake the battalions. Him Lycurgus slew by guile and nowise by might, in a narrow way, where his mace of iron saved him not from destruction. For ere that might be Lycurgus came upon him at unawares [145] and pierced him through the middle with his spear, and backward was he hurled upon the earth; and Lycurgus despoiled him of the armour that brazen Ares had given him. This armour he thereafter wore himself amid the turmoil of Ares, but when Lycurgus grew old within his halls [150] he gave it to Ereuthalion, his dear squire, to wear. And wearing this armour did Ereuthalion challenge all the bravest; but they trembled sore and were afraid, nor had any man courage to abide him. But me did my enduring heart set on to battle with him in my hardihood, though in years I was youngest of all. So fought I with him, and Athene gave me glory. [155] The tallest was he and the strongest man that ever I slew: as a huge sprawling bulk he lay stretched this way and that. Would I were now as young and my strength as firm, then should Hector of the flashing helm soon find one to face him. Whereas ye that are chieftains of the whole host of the Achaeans, [160] even ye are not minded with a ready heart to meet Hector face to face.”

The parallel is drawn out by M.L. West in The East Face of Helicon. West Asiatic Elements in Greek Poetry and Myth. On p.214 under Single Combat, and then continuing on pp.369-70:

A hostile encounter between two individuals may be the main subject of a poem (as in Anzu, or the Shield of Heracles), or a local incident within a full-scale battle. Homeric battle narrative contains many such individual encounters. This is not so in Akkadian epic, because the king is the only warrior allowed to emerge as an individual. However, from poems such as Anzu, and from the heroic saga of the Old Testament, we can find parallels for a number of the motifs that Homer employs in constructing his duels.

One situation in which a single combat may appear in the context of a battle between armies is that where a challenge is issued for a volunteer to take on the enemy's champion. Sometimes there is a prior agreement that this will settle the outcome of the war. The duel between Paris and Menelaus in the third book of the Iliad is conducted on these terms. That between Hector and Ajax in the seventh book is not, as Homer could hardly use the motif a second time. In other respects, however, this latter duel is an excellent specimen of the genre. With it we may compare the story of David and Goliath, which resembles it not just in its general concept but in the nature and sequence of its constituent elements:

1 Sam. 17	Iliad 7
Goliath issues challenge (4-10//23)	Hector issues challenge (67-91)
Dismayed reaction (11//24)	Dismayed reaction (92-3)
David will fight him (32)	Ajax will fight him (191-9)
David arms (38-40)	Ajax arms (206)
Speeches of the combatants (43-7)	Speeches of the combatants (225-43)
The fight begins (48)	The fight begins (244)
Goliath laid low with stone (49-50)	Hector laid low with stone (268-72)

Within the narrative of the duel of Hector and Ajax the poet has set a reminiscence by Nestor of a similar story in which he himself played the hero's part. Here the parallels with David and Goliath are all the more noticeable, as the defeated warrior Ereuthalion, is characterized as a giant (155 f), and he wielded a distinctive weapon, an iron club (141), which recalls the massive iron head of Goliath's spear (1 Sam. 17. 7). He came forward in his armor and challenged all the opposing heroes to fight him. The rest were afraid, but Nestor - the youngest of Neleus' twelve sons - as David is the youngest of Jesse's eight - accepted the challenge and, with divine support, killed the man and left him sprawled on the ground. ... The event is located at a river Iardanos (135, contradicting 133), which later commentators could not find; this has been suspected* of reflecting an original which referred to the Jordan (Heb. Yarden). *Another river Iardanos is located in Crete (Od. 3. 292) and another in Lydia (St. Byz.). The name is easily but perhaps falsely derived from W[est] Semitic yrd 'descend'; it may rather be pre-Semitic.

The Nestor-David parallel can be extended in that Nestor is called ἡδυεπής in the Iliad, that is, sweet-speaking, a description also given to the Muses and to singers. That extension of the parallel is my own doing.

So, we have a parallel between David and Goliath in Greek mythology: Nestor and Ereuthalion. Now we must ask, is there a connection between Hestia and Nestor? Sadly, not one that I can find. The best I can offer is that at the so-called "Palace of Nestor" is a Mycenaean hearth in a round shape, similar to the round Temple of Vesta (Miller & Miller, The Prytaneion: Its Function and Architectural Form p.25). Recall that I mentioned that Vesta is considered by Ovid to be the Earth itself; I hoped that I could tell you that Ereuthalion is a child of Gaia, like the other famous giants in Greek mythology (called Titans), but sadly that isn't the case. He's just a normal enviously-tall-and-strong man. This is where our trail runs cold, as I told you at the beginning it likely would.

Thanks for playing and I hope this has been informative!