(Tsume no aka wo senjite nomu;
“to boil and drink fingernail-dirt”)
To follow a good example that someone has set. To learn from someone who excels, by doing whatever it takes to get any positive effect at all. Often referenced as something that you want to make someone do so that they could be more like someone else, whom you view as a role-model.
We begin three characters in, with the noun 垢 (aka), “dirt.” The associative particle の (no) connects this to, and allows it to be modified by, the noun 爪 (tsume), “[finger]nail.” The resulting noun phrase is marked as the direct object of a verb by the particle を (wo), and the verb performed on it is a compound comprising 煎じる (senjiru), “to boil,” “to infuse,” in conjunctive form and the verb 飲む (nomu), “to drink,” in conclusive form.
This is not something people actually did or do; rather, it’s playing off of a sort of homeopathic version of sympathetic magic, akin to the idea that someone’s greatness might “rub off” on you simply from shaking hands, except turned into an infusion like a perverse herbal tea.
Note that the phrase 爪の垢ほど (tsume no aka hodo, “as much as the dirt under one’s nails”) is an idiom referring to a very small, even trivial, amount of something. This kotowaza borrows that sense, as well as using the literal image of fingernail crud as a thing you could infuse into a medicinal drink. The use of something so minuscule and dirty is for emphasis: you want the person to get even a tiny bit better than they are.
Attempting to contract the phrase as 爪の垢を飲む (getting rid of the boiling step) or 爪を煎じて飲む (using the nail directly instead of the dirt under it) is considered an error.
(“Kono gakusei wa kanpeki na kaitou no shukudai wo kanarazu jikan doori ni teishutsu suru kara, sono tsume no aka wo senjite minna ni nomasete mitai na.”)
[“This student turns in perfect homework, precisely on time, without fail. I wish I could make all of them even just a fraction more like that.”]