Know Your Phrase has this to say,
This phrase's origin is not clear. However, it may have something to
do with the maintenance involved in keeping a musical instrument in
good condition. Indeed, instruments like guitars, flutes, drums and
others require a level of care to keep them in good shape and
For example, let's take a look at fiddles. These typically refer to
stringed instruments, such as a violin (there is one portrayed in the
picture above). In order to help a violin remain in a working state,
its strings must be replaced if they break, tiny pegs need to be kept
tightened, and it should be cleaned every now and then to prevent dust
buildup. This sort of maintenance keeps the violin healthy or "fit,"
so to speak.
So at some point, it seems a person's health started to be compared to
a well-maintained fiddle, though why this musical instrument was
chosen out of all the others, that I do not know.
Anyhow, this saying goes back to at least the early 17th century. It's
written in a book entitled English-men for my Money, by Haughton
William in the year 1616. There's a part from it that reads:
"This is excellent ynfayth, as fit as a fiddle."
That means this expression is over 400 years old, and it could very
well be much older.
I agree with @user307254's comment about alliteration.