Had to do some digging for this one.
First off, some informed speculation:
A, of course, is the first letter in the alphabet, so its as good a place as any to define a standard. Plus, as you say, it is one of the open strings in common on all string instruments (along with D and G). The A above middle C is also a relatively centrally-located pitch, accessible to a range instruments. An octave lower would have been inaccessible to flutes and oboes (and not an open string).
To expand on this: String instruments are relatively easy to retune, but also relatively quick to go out of tune (especially when strung with gut strings), so they should tune to the more stable wind instruments. The oboe was one of the first wind instruments to be regularly added to the orchestra (in the Baroque period), and, for whatever reason, it is now used as the pitch standard (there's another good question). Oboes can't hit G3, therefore, we have the following set of constraints:
- Open string on violin: G3, D4, A4, E5
- Also an open string on viola & cello(8vb): G3, D4, A4 (no high E string)
- Also available on the oboe: D4, A4 (no low G)
- In a "comfortable" range on the oboe: A4(*)
(*) I am not an oboist, but I'm sort of assuming here that, at least on early instruments, the D4, which was the second-to-lowest note on the oboe, might have been a bit harsh sounding. I found a quote from Berlioz' Treatise on Orchestration which agrees with this notion: "The lower notes of the oboe, which sound ugly when exposed, may be suitable in certain harmonies of an eerie and sorrowful character..."
At any rate, I think the answer to your question will lie in discovering why the oboe is used as the reference pitch. Presumably, at least in the early Baroque, it would have been the continuo harpsichord (or organ) providing this reference, so maybe this shift coincided with the disappearance of continuo in the Classical era?
Some additional historical information that may be relevant:
In the early 6th century, when Boethius lists all the note names by letter, he starts (unsurprisingly!) with A as the lowest pitch. This presumably set a precedent for using A as a reference pitch.
In 1711, British trumpeter John Shore invented the tuning fork which allowed accurate tuning to a reference pitch. He even gave one to Handel, with a pitch of C=512Hz, corresponding to your scientific pitch.
In 1834, Johann Heinrich Scheibler, a silk manufacturer and acoustics researcher, invented a "tonometer" for accurately measuring pitches, based on an array of fifty-two tuning forks, spanning a range of pitches fro A3 to A4. Why did he choose A? I don't know! But this device allowed him to study a range of tunings in use at the time. He describes his tonometer in detail in Der Physikalische Und Musikalische Tonmesser, which is apparently available on amazon, if you fancy original sources and can read German.
Based on his studies, he recommended A4=440 to the Congress of Physicists in Stuttgart in 1834, which they accepted. The Streicher piano company also adopted this standard, manufacturing pianos with "440" stamped on the label. This wasn't the first attempt at standardized pitch, but due to Scheibler's tonometer, it was likely the most accurately measured standard up to that point. Nor would it be the last word in tuning; the French government passed a law on Feb. 16, 1859 requiring A=435, or "Diapason Normal" which would become widely adopted internationally.
For more reading: