It was surprisingly difficult (to me at least) to find a nice classical equivalent for the modern pair: problem-solution. In Late Latin we can find problema solvere but the term problema seems to be more suited for a technical/a riddle rather than a trouble/difficulty.
For "a way out" (as a solution) there are several alternatives like exitus and via, but I could not find the corresponding term for problem. It is as if the authors simply avoid the (abstract) term as in: inveniam viam aut factiam or difficilius est exitum quam principium invenire; I am almost prompted to also avoid the term problem by using the dative, like: for everything a solution. However, another problem with exitus and via is they are relatively broad in meaning. Another option is solutio though I'm not sure it is actually better.
Tying to inquire this problem from problem yielded the options: difficultas and negotium; from those two I would pick negotium since difficultas might be little too strong and usually implies pain and suffering - not so apt to describe a situation as described in the post of missing the right hammer; in that respect negotium seems quite right as it usually implies an issue/occupation/trouble. Meissner brings negotium expedire as "to arrange, settle a matter". I think it is a good match for "solving a problem/issue"; the problem now is that I didn't find the noun for expedire. Doing a little corpus research on negotium I finally found what was previously escaped me: magnum negotium, cuius exitum non extimesco (A big problem whose solution I fear not; Cicero).
Unfortunately, the sense of exitus in that passage is rather result/outcome/conclusion and not "a solution" as one translator has it: "there is a serious business on hand, the result of which I do not dread". But I say that this distinction is really not that big as it seems to us English speakers; the idea is a way out; Lewis and Short dictionary specifically gives "A means, method, way, device, solution of a difficulty"; i.e., the way. I believe we can add a verb like reperio or invenio to exitus in order to clarify its specific sense as some authors indeed did.
Now, the requested bold part "for a specific problem, a specific solution" calls a known pattern in Latin Suum cuique , Let me thus suggest:
cuique [proprio] negotio suum [proprium] exitum quo expeditur; huic suum. (cf. Suam cuique sponsam, mihi meam)
Every problem has its own solution (by which it is resolved); Also this has its own.
- I put proprium as optional as I think it is little redundant here, but it gives the emphasis on the specific part.
- quo expeditur can theoretically be omitted, but I found this as an important addition to clarify the meaning of exitum towards solution rather than an outcome.
- I'm still not so happy with that. Maybe it sounds too dramatic. Maybe solution, via or remedium instead of exitus would work better.
Trying to seek validation to my suggestion, I saw indeed the pair (felix) exitus - negotuium shows up and not rare, in some mediaeval texts - but it is very hard to judge the exact translation without deep reading of those texts; However, as far as I could tell, it is more of the Cicero sense above.
Then, I used luchonacho's "usual trick", and sought a solution to this problem in the Vatican texts. From what I was able to gather, there is no one formula that was used to signify a solution (maybe even different translators?), whereas for problem the translators were comfortable using quaestio (as it was more in the sense of affair/issue as "the issues of humanity" where I think quaestio makes more sense in the case here. But maybe I'm wrong here, and quaestio is good for the problem of the hammer). At any case here are some formulas (emph. mine):
"We approach the subject with confidence, and in the exercise of the rights which manifestly appertain to Vs, for no practical solution of this question will be found apart from the counsel of religion and of the Church." (source)
«Confidenter ad argumentum aggredimur ac plane iure Nostro, propterea quod causa agitur ea, cuius exitus probabilis quidem nullus, nisi advocata religione Ecclesiaque, reperietur». (source)
While it is true that many Christians understand the moral teaching of the Gospel differently from Catholics, and do not accept the same solutions to the more difficult problems of modern society, (source)
Quod si inter Christianos multi non semper eadem ratione atque Catholici Evangelium in re morali intelligunt neque easdem solutiones difficiliorum hodiernae societatis quaestionum admittunt, (source)
it is for the public authorities "to seek a solution to these questions, with the active involvement of individual citizens and social groups." (source)
ad publicas auctoritates pertinet operam dare his quaestionibus dissolvendis, rem operose communicantibus civibus ac socialibus coetibus. (source)