We know from the Book of Mazarbul that shortly before the downfall of his newly established kingdom, Balin went to look in the Mirrormere.
Is there any canonical reference to what he saw?
When the Fellowship is in Khazad-dûm (Moria), Gimli chants a poem about the original Durin. It tells how he first looked in the Mirrormere and saw a crown of stars about the reflection of his head:
He named the nameless hills and dells;
He drank from yet untasted wells;
He stooped and looked in Mirrormere,
And saw a crown of stars appear,
As gems upon a silver thread,
Above the shadow of his head.
. . . .
But still the sunken stars appear
In dark and windless Mirrormere;
There lies his crown in water deep,
Till Durin wakes again from sleep.
The Lord of the Rings Book Two, Chapter 4: A Journey in the Dark
Page 316-7 (Houghton Mifflin Harcourt; Single Volume 50th Anniversary Edition)
The place is still revered by Durin's folk as we see from Gimli's actions when he approaches it.
‘That is Durin’s Stone!’ cried Gimli. ‘I cannot pass without turning aside for a moment to look at the wonder of the dale!’
. . . .
Beside the standing stone Gimli halted and looked up. It was cracked and weather-worn, and the faint runes upon its side could not be read. ‘This pillar marks the spot where Durin first looked in the Mirrormere,’ said the dwarf. ‘Let us look ourselves once, ere we go!’
They stooped over the dark water. At first they could see nothing. Then slowly they saw the forms of the encircling mountains mirrored in a profound blue, and the peaks were like plumes of white flame above them; beyond there was a space of sky. There like jewels sunk in the deep shone glinting stars, though sunlight was in the sky above. Of their own stooping forms no shadow could be seen.
‘O Kheled-zâram fair and wonderful!’ said Gimli. ‘There lies the Crown of Durin till he wakes. Farewell!’ He bowed, and turned away, and hastened back up the greensward to the road again.
The Lord of the Rings Book Two, Chapter 6: Lothlórien
Page 334 (Houghton Mifflin Harcourt; Single Volume 50th Anniversary Edition)
I think we can conclude that Balin, like Gimli, revered the Mirrormere and liked to look in it. When he did that, he would see what Durin saw before him, and Gimli saw after him: the crown of stars. Presumably this would remind him of the past greatness of his people and of Khazad-dûm.