What is the process for translating modern words into Latin? For example, Latin Wikipedia has entries for tablet computatrum tabulare and smartphone sophophonum. Is there any particular process for coming up with these names, or do users of contemporary Latin come up with individual translations and use it until one of these "catches on"?
Basically, your second option is correct. There is the Vatican's Lexicon Recentis Latínitátis, referred to in the comment, but its approach to Latin is very idiosyncratic; debates over neologisms often get incredibly acrimonious, if you can believe it, but there are (VERY roughly) two schools: the "Ciceronian," in which making up words for concepts that didn't exist in ancient Rome is to be avoided whenever possible, and the "Erasmian," in which new concepts call for new ideas. ("Ciceronian" and "Erasmian" are my names for them, though they're not entirely accurate; Cicero made up words all the time, and Erasmus has a whole screed about how people who think you can't use any Latin Cicero didn't use are idiots.)
The Vatican is of the more "Ciceronian" school, which gives you, unfortunately, things like "immódica medicamenti stupefactívi iniéctió" as the official term for "overdose." Other than the Vatican, actually, I'm not aware of anybody who takes this approach to neologisms.
(Actually, now that I think of it, even the Vatican doesn't take this approach, or not always: The Diary of a Wimpy Kid was translated into Latin (Commentarií dé Ineptó Pueró) by one of the Pope's Latinists (Monsignor Daniel Gallagher), and it definitely doesn't use the Lexicon Recentis Latínitátis.
So basically, somebody comes up with a word and it catches on. "World-wide web," for example, is widely known as "Téla Totíus Terrae," and I have to imagine that somebody came up with that and couldn't stop giggling, and it's so great (ttt) that there's no reason to come up with anything else. "Computátrum" is the generally used word for "computer" (though the Vatican prefers "ínstrúmentum computátrálium").
For what it's worth, David Morgan of Furman University put together a lexicon of Neo-Latin words that already exist that can be used for modern concepts. Careful, though, because the Lexicon Morganianum is in two parts, the adumbratio and the silva, and people tend not to realize this. Patrick Owens, the lexicon's current curator, explains the difference:
The Morgan-Owens Latin Lexicon is found under the link entitled "adumbratio" (see link below). The link entitled "silva" is a mere collection of other modern Latin dictionaries. If a term appears in the "silva" but not in the "adumbratio" it is because it has been intentionally rejected on philological grounds.
TL;DR: People come up with stuff and whatever sticks, sticks.