A thistle is a plant which goes to seed with several "downy" seeds. If allowed to go to seed, the thistle will spread these seeds far and wide, as the down allows them to be blown aloft in every direction. The poem is often attributed to Clement Moore but is likely actually the work of Henry Livingston Jr. Both men lived in the colonial America, where the thistle is abundant, proving a nuisance if allowed to seed into fields.

The reindeer would be able to cover vast distances, both quickly and silently. When the reindeer are aloft, they would be as difficult to spot as thistle seeds.

However, I believe the primary reason the line ends on "thistle" is to rhyme with "whistle."

It is very good practice whenever you encounter a word that you are not sure of the definition, to locate that word in the poem or text you are studying and read around it, both before and after, to see if there are any contextual clues that you can use to deduce the meaning of that word. This is a valuable strategy that can be used to increase you knowledge of vocabulary. Let us see how this word fits in to the poem as a whole:

He sprang to his sleigh, to his team gave a whistle,
And away they all flew like the down of a thistle.

We can see that the reference to "thistle" is in a simile that the author uses to compare the sleigh and the reindeer to the "down of a thistle." Unfortunately, because this phrase comes as the simile, contextual clues won't help us a lot, which means we need to look up the word if we are unsure of its meaning. A thistle is a herbacious plant with a prickly stem and purple flowers.