The question is interesting though perhaps not strictly "research-level". Terminology in mathematics develops a bit haphazardly, and sometimes things get misleading names. In this case the work of Specht around 1935 did place the representations of symmetric groups in the then-modern setting of module theory. But the notion of "Specht module" seems to have emerged around 1970 in the rapidly developing work on representations of the groups in prime characteristic. Work of M.H. Peel and especially work of Gordon James popularized the notion. In particular, James exploited the fact that the characteristic 0 Specht modules have a fairly natural reduction mod $p$ for any prime $p$. This is somewhat analogous to the algebraic group situation, where "Weyl modules" come by such reductions and then have a unique distinguished composition factor.
By now the literature on symmetric group representations in prime characteristic is quite extensive, with the term "Specht module" being ubiquitous. In the classical characteristic 0 theory, no such language is usually needed. Anyway, one might make a case for the terminology "James module" here, but it's too late for that. Specht himself had no special influence on the modular theory.
I think that understanding what can be a "motivation" for Beilinson-Drinfeld work, at least several things should be kept in mind.
1) V. Drinfeld made principal contributions for functional fields case of Geometric Langlands - he established it for GL(2), as well as, creating basic constructions which lead Lafforgue to establish GL(n) case - he introduced "schutakas", and a way to use them in order to get Langlands correspondence.
( Somewhat curious, that his contribution to this field dated back to 1979, and since that time he seems to moved his principal interests to quantum groups and "math. physics". And it seems that it is kind of astonishing that work on geometric Langlands over complex numbers unified both of his interestes - "math. phys" and Langlands.)
2) A. Beilinson (with J. Bernstein) introduced so-called "Beilison-Bernstein localization" in their way to prove Kazhdan-Lusztig conjectures on Verma modules and intersection cohomology. This construction plays an important role for "understanding-motivating" Beilinson-Drinfeld work.
It defines a correspondence between D-modules on flag variety and certain representations of the semisimple Lie algebra.
( In Beilinson-Drinfeld story: similar D-modules apppear as "Hitchin eigensheaves", flag manifold is substituted by moduli space of vector bundles, semi-simple Lie algebra substituted by affine Lie algebra).
3) Hitchin's paper on integrable system appeared in 1987, and pay attention that it appeared in special issued of Duke journal dedicated to 50-th anniversary of Yu.I. Manin,
who was the teacher of both V.Drinfeld and A.Beilinson. So it was of course known to them.
From the position of our nowdays knowledge it is easy to understand the relevance
of Beilionson-Bernstein (BB) localization to Hitchin's paper: roughly speaking
the center of universal enveloping of affine Lie algebra on the critical level
which play crucial role in BB-localization gives exactly the quantum version of the Hitchin integrable system.
However, I think in late 1980-ies, probably no one except Beilinson and Drinfeld were able to see through the clouds. (Since the center itself was not actually proved to exist in 1987, and many many other things were not yet known).
Concerning the more detailed version of the question.
Why are moduli spaces of stable Higgs bundles (arising in Hitchin systems) on Riemann surfaces the right place to look at and not just moduli spaces of stable vector bundles on Riemann surfaces ?
I am not sure I fully understand the point to make stress on the moduli space of Higgs bundles vs. moduli space of vector bundles.
In my undestanding one should stress on the following things.
1) Moduli space of Higgs bundles is (modula details) COTANGENT BUNDLE to moduli space of vector bundles. Contangent bundles to manifold appears naturally when you speak about D-modules or "quantization" - it is "classical phase space" in quantization story or the space where characteristic manifold of D-modules lives.
2) What is surpsising that in functional field case developped by V.Drinfeld in late 70-ies, he needs to work with moduli space of "schtukas" - which are vector bundles plus additional structures related to Frobenius, while Beilinson-Drinfeld story is more simple in that respect - you do not need "schtukas" and you can actually work with vector bundles.
Best Answer
The term was coined by one Michael de Finkelberg during his visit to Croatia. The word is indeed Croatian and means ``flag''. I was happy to have a Croatian word in mathematics. The strategy of giving a new notion an old name but in a different language is not perfect.