That tale about Elros and Elrond appears in a letter dated 1958, and the idea seems quite connected to Elrond meaning *Elf of the Cave.
Not long after however, Elrond is said to mean 'Star-dome' (1959-60, Quendi and Eldar), and in The Shibboleth of Feanor (1968 or later) it was said that the names Elros and Elrond:
'were formed to recall the name of their mother Elwing' with Elros meaning 'Starlit foam'. In The Problem of ROS (1968 or later):
'Now Elrond was a word for the firmament, the starry dome as it appeared like a roof to Arda; and it was given by Elwing in memory of the great Hall of the throne of Elwe in the midst of his stronghold Menegroth that was called the Menelrond,...'
And in letter 345 (1972) Elrond meant 'The vault of stars'.
If a meaning 'Elf of the Cave' is out, as it appears, my
guess would be that so too is the story that goes with it. Tolkien could have retained the detail of course, or something similar regarding the young brothers, but with JRRT -- as stories might spring from names -- I thought I would at least point out my annoying, somewhat off topic guess.
