Just as the topic says, how did the name "analysis" come to denote the specific mathematical branch dealing with limits and stuff? The term "analysis" seems very generic compared to the words for the other two main branches, "algebra" and "geometry", which do not seem to have other unrelated meanings.
There is a tradition on early modern mathematics regarding the usage of the term analysis :
François Viète, Isagoge in Artem Analyticem (Introduction to the Analytic Art), Tours, 1591 (several successive editions and translations);
Thomas Harriot, Artis Analyticae Praxis, London, 1631.
The background is the "rediscovery" of ancient Greek mathematics and, in particular of Pappus of Alexandria, (c.A.D. 290 – c.350) and his main work in eight books titled Synagoge or Collection, which Book VII explains the terms analysis and synthesis, and the distinction between theorem and problem.
See Henk Bos, Redefining Geometrical Exactness. Descartes' Transformation of the Early Modern Concept of Construction (2001), page :
Two kinds of analysis were distinguished in early modern geometry: the classical and the algebraic. The former method was known from examples in classical mathematical texts in which the constructions of problems were preceded by an argument referred to as "analysis;" in those cases the constructions were called "synthesis".
Reference to Pappus' problems is also found into René Descartes' La Géométrie (1637).
The two main line to be understood are :
analysis as a "method" to solve problem
analysis as the technique of treating geometrical problems with algebraic methods.
Both, I think, are "involved" into the use of analysis to name the new method introduced by Newton and Leibniz.
You can see :
Jaakko Hintikka & U.Remes, The Method of Analysis: Its Geometrical Origin and Its General Significance (1974)
Michael Otte & Marco Panza, Analysis and Synthesis in Mathematics: History and Philosophy (1997).
The first occurrence (1696) of the term "analysis" in the sense of the mathematical discipline extending calculus occurs in the title of l'Hopital's work Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes. The term had been used earlier as part of a dichotomy analysis/synthesis, for example in Fermat. However l'Hopital was the first to use the term to describe the new science being created by Leibniz and others in the 17th century.