"Light Brigade" refers to a specific brigade
The Charge of the Light Brigade was a real-world event. Quoting from Wikipedia:
The Charge of the Light Brigade was a failed military action involving the British light cavalry led by Lord Cardigan against Russian forces during the Battle of Balaclava on 25 October 1854 in the Crimean War.
The poem was about this event:
The events were the subject of Alfred, Lord Tennyson's narrative poem "The Charge of the Light Brigade" (1854), published just six weeks after the event.
Since the brigade had light cavalry, it is the Light Brigade. To indicate that a specific, special brigade is being talked about, a defining trait ("light cavalry") is capitalized and used as a descriptor. This is a general pattern used in other situations. A book club that contained mainly old ladies might be called the Old Ladies Book Club. A section of a zoo devoted to birds might be called the Bird Section. The capitalization is used to indicate specialness.
That's the question in the body. Now to address the title. One note before I do: you've misquoted the poem. Tennyson uses British spellings for words, and as this Poetry Foundation copy of the poem shows, he spells it "honour".
Quoting Merriam-Webster, definition 1b:
to give special recognition to : to confer honor on
Tennyson wants the Light Brigade to be honored, or given special recognition. In other words he wants acts of remembrance for the Light Brigade; he wants people to talk about them with importance and not simply forget that the event occurred.
Therefore "Honour the Light Brigade" means "Confer special recognition on this specific light-cavalry brigade"