I came across your intriguing question and I felt compelled to 'tuck in'.
In Ezekiel 2:6
And thou, son of man, be not afraid of them, neither be afraid of their words, though briers and thorns be with thee, and thou dost dwell among scorpions: be not afraid of their words, nor be dismayed at their looks, though they be a rebellious house.
The briers, the thorns and the scorpions are the rebellious. It's same figure of speech in
Trees are symbolic for people when their deeds are the subject matter;
6 And all the men of Shechem gathered together, and all the house of Millo, and went, and made Abimelech king, by the plain of the pillar that was in Shechem.
7 And when they told it to Jotham, he went and stood in the top of mount Gerizim, and lifted up his voice, and cried, and said unto them, Hearken unto me, ye men of Shechem, that God may hearken unto you.
8 The trees went forth on a time to anoint a king over them; and they said unto the olive tree, Reign thou over us.
9 But the olive tree said unto them, Should I leave my fatness, wherewith by me they honour God and man, and go to be promoted over the trees?
10 And the trees said to the fig tree, Come thou, and reign over us.
16 Now therefore, if ye have done truly and sincerely, in that ye have made Abimelech king, and if ye have dealt well with Jerubbaal and his house, and have done unto him according to the deserving of his hands;
From the above verses, the trees depicted Israel;
22 Then the men of Israel said to Gideon, "Rule over us, both you and your son, also your son's son, for you have delivered us from the hand of Midian."
23 But Gideon said to them, "I will not rule over you, nor shall my son rule over you; the LORD shall rule over you."
The olive, the vine, the fig, etc denoted more noble individuals than Abimelech, his own slain brothers, himself as the bramble.
In Isaiah 55:13 God signifies by a thorn and a brier, a type of people that walks in specific rebellious ways relative to another type of people, denoted as the fir and the myrtle, that walk in righteous ways.
The latter category are those for a name to the LORD, a name that will not be cut-off.
Cut-off is a term specific to judgement upon wicked spirits.
Note below that even when an idolater has already been killed, or in the case when people have no will to kill him, God says he will 'cut him off'; what Matthew 10:28 properly summarises.
2 Again, thou shalt say to the children of Israel, Whosoever he be of the children of Israel, or of the strangers that sojourn in Israel, that giveth any of his seed unto Molech; he shall surely be put to death: the people of the land shall stone him with stones.
3 And I will set my face against that man, and will cut him off from among his people; because he hath given of his seed unto Molech, to defile my sanctuary, and to profane my holy name.
4 And if the people of the land do any ways hide their eyes from the man, when he giveth of his seed unto Molech, and kill him not:
5 Then I will set my face against that man, and against his family, and will cut him off, and all that go a whoring after him, to commit whoredom with Molech, from among their people.
A name here denotes essence, what makes an angel to be an angel, for example or a human being to be a human. That is the name meant here.
That is to say, concerning a people with a wicked nature that plagued Israel, and this wickedness is also categorised by the presence of a thorn and a brier, the LORD promises to 'form' a people with a righteous nature in the symbols of fir and myrtle.
This has always been God's plan for Israel;
This new nature or essence is the name that will not be cut-off.
A sign is contextual, denoting a servant who convenes, in Isaiah 66:19 & Matthew 24:30, it's messiah.
But in this case of Isaiah 55 the sing refers to a permanent feature of reference, as a reminder, as meant in;
Also I gave them My sabbaths to be a sign between Me and them, that they might know that I am the LORD who sanctifies them.
So, whereas other 'names' can be 'cut-off', for example, the famous name of 666 having which sends one into the infernal lake, the essence or name that these fir trees and myrtle trees become, from above, is forever.
The essence or the new nature of the righteous people, these denoted as the fir and the myrtle, that is the referent of 'it shall be to the LORD for a name'. Simply put, it's His 'name', Isaiah 56:5 and Isaiah 43:7, forexample.