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I don't remember the history too well, but the answers above perfectly fit one of my favorite quotes from V.I. Arnold on this very question that illustrates the gulf between (1) axiomatic training and (2) hands-on approach.

1. there's the obvious stupid answer: on an affine scheme, restriction to distinguished open sets corresponds to localization of the ring. It seems rather clear that localization is a good name for this, especially since you can look at smaller and smaller open sets around a point. (Ilya Grigoriev)

2. If $M$ is a manifold and $x\in M$ , then the ring of smooth germs in $x$ is canonically isomorphic to the localization […] Geometric localization is expressed algebraically in introducing inverses for functions for which it makes sense, thus you can call it localization. (Martin Brandenburg)

Студенты высшей нормальной школы в Париже спросили меня: "Почему вы называете кольцо формальных степенных рядов локальным? Разве оно удовлетворяет аксиомам локального кольца?" Для неспециалистов поясню, что заданный вопрос аналогичен вопросу "Почему вы называете окружность коническим сечением?" Это были лучшие студенты-математики Франции. По-видимому, какой-то преступный алгебраист обучил их аксиомам колец (и даже локальных колец), не приводя ни одного примера (и, в частности, не объяснив происхождение термина "локальное").

(В.И. Арнольд, Топологические проблемы теории распространения волн, УМН, т.51, вып.1 (307), 1996, с.5)

Here is my rather literal translation:

Students from École Normale Supérieure, Paris asked me: "Why are you referring to the formal power ring as local? Does it really satisfy axioms of a local ring?" Let me remark for the non-experts that their question is analogous to the question: "Why do you call the cirlce a conic section?" Those were the best mathematics students in France. Apparently, some criminal algebraist taught them ring axioms (and even local ring axioms) without giving a single example (and, in particular, without explaining the origin of the term "local").

(V.I. Arnold, Topological problems in the theory of wave propagation, Russian Math Surveys, 51:1, 1996)