From the Japanese version of Harry Potter:
「チャーリーにまたふくろう便を送る暇はないし、ノーバートを何とかする最後のチャンスだし。」
There's no time to send another owl to Charlie. This is the last chance to somehow do Norbert.
Norbert is a dragon that the speaker has arranged, with Charlie, to get rid of.
I know that する has several meanings but I can't find one that makes sense in this context. The original text says "...chance to get rid of Norbert". How does する translate to "get rid of" here?
There might be room for a more complete answer, but since it seems they did the trick, might as well turn my comments into one:
<(source of) issue>を何とかする = "do something about <(source of) issue>" [in order to resolve it]
Examples from media for 何とかする. It might also help to check out some examples in other forms as well (such as 何とかしないと・何とかして・何とかしろ, for some common ones).
I figured out how to situate this within a larger framework. The relevant broader concept here is the use of する and なる to say that things will "work out".
Canonically, you might think of なる as a verb meaning "to become", but sometimes it has a meaning of things "working out", or of a situation "turning out alright". Consider, for instance, the ～なければならない "must do" construction. You could decompose this to ～なければ + ならない ("if I don't do such-and-such, then things won't turn out well" → "I need to do it") (though to what degree people actually think about it like that is another question, since it's a fairly set grammaticalised expression, but that's beside the point).
Likewise, apart from 何とかする, there's its intransitive/non-volitional counterpart 何とかなる. So, in the kind of context where なる refers to things sort of naturally or circumstantially turning out alright (with the speaker lacking strong, direct control over the outcome), する refers to some agent taking matters into their hands and actively making things fine.
Consider also how する and なる complement each other in なる's more canonical usage of simply denoting change or "transformation":
(Notable special cases like the 気にする・気になる pair might come to mind.)
How does する translate to "get rid of" here?
It doesn't. Contextually, the suru-verb compound 何とかする does. A more literal translation would be "do something (about)". We can figure this out by e.g. looking up 何とか in a dictionary.
The translation isn't literal (using some form of [捨]{す}てる, I guess?) because the English isn't literal. Norbert isn't a possession to be discarded; "getting rid of" him doesn't mean abandoning him, but rather taking some unspecified action (perhaps killing; but merely preoccupying him with something else would qualify - I don't know the story that well) that will prevent him from interfering in some future situation.
We know that we have to think of the entire compound 何とかする as the predicate here, because か isn't a case-marking particle. We can interpret 何とか contextually as adverbial, but that would be less common with する (but yes, it would mean "somehow do") and obviously doesn't make sense here (because the を-marked part isn't an action).
