I'm not sure that I'm historically accurate, but that is how I always thought about algebraic nomenclature.

1) Group actually comes from group of substitutions. I guess that Galois could have introduced any other word, like "set" of substitutions or "flock" of transformations. Set theory was not yet established, so I guess a collection of functions could be called 'group', 'set' and so on according to the taste.

2) For field, I guess it comes from the meaning of field as "sphere", "subject", "area". It makes sense that such a word could come in talking about "solving an equation in the real field" rather than "solving an equation in the complex field". Then the concept of an abstract field could have followed.

3) Ring comes from "Zahlring", ring of numbers. This, as far as I know, is a terminology due to Dedekind. He was actually working with number rings, of the form $\mathbb{Z}[\alpha]$, where $\alpha$ is integral over $\mathbb{Z}$. So for some $n$, $\alpha^n$ can be expressed in terms of lower powers of $\alpha$; in some sense the components of the basis of $\mathbb{Z}[\alpha]$ over $\mathbb{Z}$ cycle, although this is accurate only when $\alpha$ is a root of unity. Hence the name ring of numbers.

4) Ideal is easy. When Dedekind realized that in a ring like $\mathbb{Z}[\sqrt{-5}]$ unique factorization does not hold, he searched for a substitute. He then realized could restore unique factorization allowing something more general than elements, the ideals. These are now called this way since he thought of them as "ideal elements" of the ring. useful to restore unique factorization. It is a fortunate coincidence that indeed for the rings he was working with (which are now called Dedekind rings), unique factorization for ideals actually holds.

5) Idéle has the same origin, being the contraction of the French "idéal élement", although the wording is inverted with respect to French use.

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