It is always tempting to try to find additional layers of meaning in Tolkien's
works, which are so densely written and brimming with linguistic and mythic
references. But here I think you are going down a blind alley.
You raise two linked questions (how clever had there been three!): one, does
the character-name of "Boromir" in Lord of the Rings refer to or derive in any
way from "Borromean"? and two, could this be applied as a "symbol" to the end
of the first book, where the Fellowship is broken because Boromir dies, i.e.,
one element is removed from a combination that thereby falls apart?
Consider a couple of ways to approach the question.
First, Tolkien would maintain that a double 'r' is quite different from a
single 'r' in words. He would also note that a missing consonant at the end of
a word is a significant difference: Boromir ends in 'r', Borromeo ends with a
vowel. On these grounds alone, I think we might say they are quite different
words, sound quite different when spoken properly, and really have nothing to
do with each other.
Second, Tolkien does give an etymology of Boromir within the system of his own
invented languages. In the Silmarillion legends (that predate Lord of the
Rings), Boromir is a man, the second son of Bor. Bor, although one of the
"swarthy men" of the East, faithfully fights with the Elves in one of the
battles against Morgoth, in contrast to his treacherous fellow chieftain Ulfand
(The Lost Road, pb edition, p. 148). Bor means "faithful" in Elvish (the name
is that given to the man by the Elves who knew him, not his own name in his own
tongue, which is not recorded). -Mir means "jewel" or "treasure", and so the
Elves called Bor's second son: Bor-omir, Treasure of the Faithful Man. (Ulf-,
as in the treacherous chieftain Ulfand, means "hideous" or "horrible" - the
Elves were not subtle in the nicknames they gave their allies and enemies.)
Boromir of the Fellowship of the Ring was invented by Tolkien after the story
of the first Boromir had been "recorded" (and a pretty scanty story it is, too:
Bor, his son Boromir, and the other "Bor-" brothers, are all killed in the
Battle of Unnumbered Tears. That's it. But they were "faithful", unlike so many
other Men whom Morgoth seduced, so the Elves remember and honor them in their
tales).
Now, Tolkien's first note on Boromir at the Council of Elrond goes: "Boromir
the (lord? Land?) of Ond. These men are besieged by wild men out of the
East..."
Ond later becomes Gondor, but the essence is there from the beginning: this
Boromir represents "good" (one might say "faithful") men, who are fighting the
Darkness and who come to the Elves for aid and advice. As editor Christopher
Tolkien observes, Boromir was "so named unhesitatingly from the start." (Return
of the Shadow, p. 398)
Thus we see that "our" Boromir's name is completely understandable as a
meaningful if unstated reference to the earlier, mythic Boromir of the
unpublished Silmarillion; and that Boromir's name is completely understandable
as a construct of Tolkien's languages, where the name represents the quality of
the person it belongs to (i.e., faithful, a treasure).
Third, Tolkien had this to say about people's attempts to explain the origin of
his names in Lord of the Rings:
I remain puzzled, and indeed sometimes irritated, by many of the
guesses at the 'sources' of the nomenclature, and theories or fancies
concerning hidden meanings. These seem to me no more than private amusements,
and as such I have no right or power to object to them, though they are, I
think, valueless for the elucidation or interpretation of my fiction....Many of
them seem to show ignorance or disregard of the clues and information which are
provided in notes, renderings, and in the Appendices....
The bulk of the nomenclature is constructed from these pre-existing languages,
and where the resulting names have analysable meanings (as is usual) these are
relevant solely to the fiction with which they are integrated. The 'source', if
any, provided solely the sound-sequence (or suggestions for its stimulus) and
its purport in the source is totally irrelevant. (Letter 297, August
1967)
The upshot of the latter paragraph is that, even had Tolkien heard the
term "Borromean" and somehow adapted it to his Elvish languages so that
Bor-omir might sound pleasingly similar, he would insist that the resulting
invented meanings of "Bor" and "Mir" take precedence in interpreting the
resulting name, rather than any possible references from the original
sound-source of "Borromean".
Fourth and lastly, the symbolism of the Borromean Rings. The three interlaced
rings as a motif symbolizing strength in combination goes back a long way, and
occurs in many cultures. The specific episode that gave them their name was a
story of Renaissance Italian family politics. The Borromeo family adopted the
triple rings as part of its family crest to show that it was related by
marriage to two other strong families, the Visconti and the Sforza. One of the
family palazzi on the Borromean Islands in Lake Maggiore is simply crawling
with representations of the device. After this, people began to refer to the
rings as "Borromean Rings". Mathematicians did not name the topological
problems involved as "Borromean" until the 1960s.
It is a wonderful iconic image, indeed. But in the context of the Lord of the
Rings, I can't imagine one associating it with the Fellowship and the breaking
that takes place at the end of the first volume. The number three has such
power, that Tolkien uses it deliberately, not accidentally. The Borromean
Three-Ring image doesn't seem to apply here, even when taken only
numerologically. There are nine in the original Fellowship (three threes), but
only eight at the dissolution at Parth Galen. There are five races in the
Fellowship, four after Gandalf is lost. Boromir is one of two Men. The loss of
Boromir results in a disassembly into three resulting storylines
(Merry/Pippin, Aragorn/Legolas/Gimli, and Frodo/Sam), not two as the Borromean
image would suggest.
Of course, there are Three Elven Rings under the sky, as the phrase
goes. But they remain in the background of the story; and it seems debatable if
the intertwining of the Borromean pattern, applied to them, really connects
with any aspect of their role in the fight against Sauron.
So in conclusion, since you asked: no, I don't think Tolkien "had this theory
in mind when he created the character Boromier." But these are just my
thoughts, as we say in
Middle-earth.

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