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.