@ClaraDíazSanchez's answer seems to cover Lewis's struggle to name this book very well. I have no additional insights on the process of choosing its name.
Instead, I come at the question from a different angle. The OP seems to take the view that "The Silver Chair" is poorly suited to be the title, dismissing the chair itself as an insignificant detail. So, "does a deeper analysis of the story put great significance on the silver chair?" Even if not, entitling the book "The Silver Chair" does. That makes the chair a symbol of prince Rilian's imprisonment by the Lady of the Green Kirtle, which is the central conflict of the entire novel.
The chair is a natural choice for that role, being a key mechanism by which the Lady maintains her power over captive Rilian. For an hour every night he is bound to that chair, supposedly to protect those around him from a violent psychotic episode that he undergoes. It turns out, however, that he is under the Lady's mind control, and as he tells the children during a period of lucidity,
Every night I am sane. If only I could get out of this enchanted chair, it would last.
Destroying the chair produces a loud, foul-smelling flash, underscoring its unnaturalness, at which Rilian again attributes the enchantment to the chair, proclaiming:
Lie there, vile engine of sorcery, [...] lest your mistress should ever use you for another victim.
Whatever inspiration led Lewis to choose "The Silver Chair" for his title, it is well suited indeed, notwithstanding that the first-time reader will not realize that until late in the story.
But what about that deeper analysis? All Lewis's fiction has significant allegorical character. The Lion, the Witch, and the Wardrobe especially so, but The Silver Chair, too. For example, many take Aslan the lion as an allegory of Jesus Christ, and Lewis himself goes even further, actually identifying Aslan with Christ:
Since Narnia is a world of Talking Beasts, I thought He [Christ] would become a Talking Beast there, as He became a man here.*
When The Silver Chair is viewed from the angle of Christian religious symbolism, it is easy to draw parallels between the talking serpent who deceived Eve into eating the forbidden fruit and the Lady / serpent who bewitched Rilian and later attempted to bewitch the children into disbelieving in everything they knew, especially Aslan. Moreover, the Lady is engaged in a campaign to invade and seize control of Narnia, Aslan's chosen kingdom in that world. If the Lady is the arch enemy of the kingdom of Aslan, then that makes her an analog of Satan.
To continue this stream of analogy, a major doctrine of Christian theology is that without Christ, human beings are trapped, enslaved, bound by the principle of sin, unable to do right by God's perfect standard. In The Silver Chair, the chair is the mechanism by which the Lady binds Rilian to her will, intending to use him for her own purposes against Narnia and Aslan. In the Christian analogic framework, this entrapment and binding to act against Aslan / Christ, combined with ultimate rescue by Aslan's agents, lends itself well to association with the concept of enslavement to sin. In that analogy, the chair itself represents sin, which is indeed a great symbolic significance.
*As quoted in Companion to Narnia: Revised Edition (2005) by Paul Ford.