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I can't seem to find anywhere where I can look up reliably the meaning and etymology of this word: finewirer. A quick search on Google gives you uses of this word in texts such as Terry Pratchett's The Color of Magic:

Brilliant constellations shone down on the Discworld. One by one the traders shuttered their shops. One by one the ganefs, thieves, finewirers, whores, illusionists, backsliders and second-story men awoke and breakfasted. Wizards went about their polydimensional affairs. Tonight saw the conjunction of two powerful planets, and already the air over the Magical Quarter was hazy with early spells.

Or Nick Kyme's Sherlock Holmes: The Legacy of Deeds:

Fearing she might be a finewirer, I recoiled and took a firm grip of my wallet, but upon seeing her and the obvious distress she was in, I realised my error.

Eric Partridge's A Dictionary of the Underworld: British and American returns the following definition:

Long-fingered thieves expert in emptying ladies' pockets

And I can see now how the meaning of the word is obvious, given that one of the entries for wirer is, according to the 5th edition of the American Heritage Dictionary:

A trapper who uses wire traps to snare game.

It seems like Victorian slang. However, I cannot find anything about the history of the word; how it came to be, and where and when it was used.

Apart from this word, how do you go about researching terms like this one?

Fine wirer Skilled pickpocket.

From the glossary of Graham Nown, Elementary, my dear Watson: Sherlock Holmes centenary; his life and times (1986) ¹

You've established the meaning of finewirer - not a straightforward thing as it is an obscure word, verging on cant. The term seems to be used most often with an explanatory gloss or, in novels, with some nod to its abstruse nature.

For example:

“I was well on my way to being a fine wirer when I gave it all up. More's the pity! But I'm doing a better work now!”

“What in the world is a fine wirer?” Flora asked.

The young man held back his head and laughed. “You two don't know nothing!...”²

Clarissa Ross, China Shadow (1974), p97

---

'E said one boy as Mr Durban caught weren't five or six at all, 'e were more like ten, an' 'e were a right thief, 'alfway ter bein' a fine wirer. That's someone as can pick a lady's pocket an' she'll never even feel it.'

'I know what a fine wirer is. ...'³

Anne Perry, Execution Dock (2001)

---

Chaff told him to watch and learn because a cracking fine wirer was going to work. She smiled at that. Did Ash even know what a fine wirer was? She wound her way into the crowd with practiced ease.⁴

Nikki McCormack, The Girl and the Clockwork Cat (2014)

As to its etymology, according to Stephen Van Dulken in Inventing the 19th Century: The Great Age of Victorian Inventions (2006), at p150:

There was a great deal of crime in Victorian London, and it has been said that the reason why so many thieves looked famished was that it was an overcrowded profession...

'Tailing' was the business of stealing handkerchiefs from back pockets. Pickpockets were dubbed 'buzzers' or 'dippers' while those who helped by obstructing pursuit were the 'stalls'. Those who specialised in stealing purses were 'finewirers' or 'maltoolers', a reference to their fondness for using (unpatented) devices to steal purses. The devices included slender blades to rip pockets, or tools which had a three-way gripping hold that could be inserted into pockets, and which cost 10 shillings (50p) a time from disreputable shopkeepers.

This makes intuitive sense, although the author might have a bias to seeing technological etymologies, given the thrust of his book, and any sources cited are not apparent from what can be seen in Google Books.

This etymology doesn't necessarily contradict the suggestion you make - that there may be a link with wire in the snare an animal (with wire) sense - although it doesn't confirm any direct link either. Wire itself seems essential to both the trapping and the 'pocket-fishing' so finewirer and wirer (in the trapping sense) might well be completely independent of each other.

Having dealt with the wire, moving on to the fine: a finewirer seems to be a cut above the average cutpurse, the most sophisticated of their profession, and the fine probably refers to their high degree of skill rather than fine-gauge wire, fine meaning:

very precise or accurate...

delicate, subtle, or sensitive in quality, perception, or discrimination...

superior in kind, quality, or appearance ⁵

(Note emphasis has been added to quotes throughout.)

Your question about research methods in general

Apart from this word, how do you go about researching terms like this one?

is probably better suited to the EL&U Meta site where it will get a better answer than I could possibly give it. For what it's worth, I found all this information with a Google Books search for finewirer.

The Green’s Dictionary of Slang has the term “fine wirer”. The term derives from “wirer”, pickpocket, first registered in ‘Ducange Anglicus’, Vulgar Tongue,1857 edition.

fine wirer (n.) (also fine worker):

the most skilful grade of pickpocket, esp. one who steals from women.

Wirer:

an expert pickpocket who uses a wire to remove objects from his victims.

Early usage examples:

1869 [UK] J. Greenwood Seven Curses of London 89: Long-fingered thieves expert in emptying ladies’ pockets – fine wirers.

1900 [US] A.H. Lewis ‘The Humming Bird’ in Sandburrs 24: Mollie’s d’ flyest fine-woiker on Byrnes’s books [...] D’ supers has to be yellow; d’ white kind don’t pay; an’ d’ rocks has to be d’ real t’ing. In d’ old day, Mollie was d’ king of d’ dips, for fair!

1903 [US] A.H. Lewis Boss 168: A mob of them Western fine-workers are likely to blow.

There is some exposition on this phenomenon of neologisms/ nonce words formed from a metaphor taking on a suffix that I found in the Russian paper by Indyushnaya AB, 2018, p.35ff.

Google-Translated in sections:

So, in English there is a combination of “fine wire”, denoting “thin wire”, to which the -er suffix has been added,

…
_{“One by one the gonophs, thieves, finewirers, whores, illusionists, backslioders and second-storey men awoke and breakfasted” [Pratchett 1983:85].}

…

According to the translation to the text he means “gentleman of fortune”, that is, a man of the underworld, living one day, constantly experiencing fate.

…

most often the occasional title exposed such objects of reality as:

- names of people on various grounds (more often on ethnic etymological, professional), for example
drugmasters (pharmacists), weatherwax (windwax, magician),

…

A complete list of occasional vocabulary with analysis is presented in the following table.

…

That table of 50 "occasionalisms," as the author calls them, may be of some benefit to the OP.

About Pratchett:

Additionally, the author has identified another 20 words that were created by the author (Terry Pratchett) "without (a) derivational model. Their meaning can only be understood through context." And goes on to believe that their etymology "is known only to Terry Pratchett himself,
since they were created according to his plan," including Alohura (goddess of lightning), Bel-Shamharoth (soul-eater), …