How did an "arm" become a "mile"?

As Peter Shor commented, the foreign expressions are all related explicitly to parts of the body:

"give him a finger, and he'll take an arm" (French),
"give him a finger and he'll take a forearm" (Spanish), and
"give him a hand and he'll take an arm" (Italian).

While "inch" may be about a thumb's-width*, it's a unit of length, so "ell" needs to be replaced with a unit of length, and "mile" has a suitable element of hyperbole. The continental expressions are parts of the body, so the hyperbole is body-related too, rather than explicitly distance-related. That is, the comparison must be within a single class.

"Yard" would certainly have a been a contender to replace an obsolete comparison, but since there is a more hyperbolic word in the same class, mile, why not use that?

*Edward II defined the inch in terms of three barley-corns laid end-to-end, so even the thumb isn't really relevant here.

I don't know where the phrase came from, but the reason for the change from "ell" to "mile" is obvious: the word ell had fallen completely out of use.

In fact, the last time I saw it used was in the early 1970's, describing this exact expression in an story from the 17th century: the Earl of Donegal was attempting to get control of an "inch", a Gaelic word for a freshwater island, and a political opponent wisecracked, "Give him an inch and he'll take an ell." After the Earl won the struggle and gained title to the inch, he rubbed it in by changing the spelling of his title to "Earl of Donegall" -- taking an l.

The story is probably apocryphal, but the county and the peerage are spelled differently for some reason.