Meaning in Context – What’s the Meaning of ‘Flannel They Run Up’?

The most important point: This is not an everyday modern usage. This passage is from a personal letter written more than a century ago. The writer uses many informal or slang constructions, but he isn't writing for us as his audience. Whoever he wrote the letter to was presumably familiar with his usages and understood perfectly well. We're also hampered by the fact that historical colloquialisms can be harder to pin down than formal usages (consider the difficulty of using this similarly informal "letter home" as a source to document the phrase "freckle past a hair"!).

Second point: The context can help us a lot. Overall, this passage is about the writer's discomfort with the lack of manners and cultivated discourse in the "wild west" and his attempts to match his peers' "rough" ways. Here is a broad paraphrase, attempting to amplify what he leaves implied and update dated usages:

I have no doubt I will do well out here despite my challenges, but you might not think the coarseness I've developed is proper for someone brought up "back East," in a more urban and sophisticated setting. The men here are rough, they grunt and growl and have no polished table manners; for instance, they guard their plates with their arms. Now I too reach past my neighbor instead of asking politely for food to be passed, and grunt, too, and shove, too, and I am almost inclined to use profanity just for the pleasure of actually communicating, saying something out loud. There is not a lot of stimulating conversation here: I don’t believe I have said more than ten words since I came to this place. I realize my attempts at profanity might be embarrassing around these "experts"; any swear words I might devise would pale next to the colorful gobbledygook they traffic in up here…

As you can see, my guess about the final sentence is that it is continues the writer's characterization of his peers' speech as inarticulate and unpolished. I'll offer some support for this reading below, but without sources that address this time and place more directly, we can't be certain. If I find such sources I'll update this answer.

"Flannel" seems to be used here to mean conversation, especially a dismissive term that suggests that the speech doesn't amount to much. It's significant that all the dictionaries I'm about to list label the usage as British. We have no indication that the letter-writer is British, so either the phrase was valid in American usage at the time, or perhaps the writer was of recent British extraction.

• Merriam-Webster: "4. British : flattering or evasive talk. Also : NONSENSE, RUBBISH"
• Dictionary.com: "British. 4b. Informal. nonsense; humbug; empty talk."
• Cambridge Dictionary: "UK informal speech containing a lot of words that is used to avoid telling the truth or answering a question, and is often intended to deceive: Leave out the flannel and answer the question!
• Lexico: British informal Bland fluent talk indulged in to avoid addressing a difficult subject or situation directly.

These definitions have the advantage that they deal negatively with speech, which is the author's context. However, his talk of men "grunting and growling" suggests that some parts of these definitions are less applicable ("flattering," "fluent"), and the main focus is "nonsense." Perhaps the letter documents a stage or offshoot in usage that emphasized the emptiness of the speech without emphasizing deception.

"Run" seems to be used here in a colloquial way that extends another usage. It could be related to "run [one's] mouth," but it seems to me it should be understood as "traffic in," perhaps extending one of the usages that means "to operate something," "to produce" especially by printing, or in particular "to transport or smuggle."

"Up" should be understood as phrasally connected to "here," not to "run": "up here."