Reference for restriction of a simple module over a splitting field to a smaller field?

I'm sure you have worked all this out, but the representation over $E$ can be realised over the extension of the prime subfield generated by the traces of the group elements (boiling down to the absence of Schur indices over finite fields, and ultimately to the fact that finite divisions rings are fields). If you assume that $E$ is that minimal field then the unique simple $FG$-module $M$ which has the given one as simple summand (after extension of scalars to $E$) has dimension $[E:F]$ times the original dimension, as George suggests, and is the sum of $[E:F]$ Galois conjugates of the original one. Furthermore, $E$ is the ring of $FG$-endomorphisms of $M$. If the field $E$ is not this minimal field, it seems less obvious to me how to realise the representation over the subfield generated by the traces, and I suppose this may be the real issue you need to resolve. Note added later: To be even more explicit in the case where $E$ is minimal, we may write $E = F[\alpha]$ where $\alpha$ is a generator for the multiplicative group of $E.$ Then $\alpha$ has minimum polynomial $m(x)$ of degree $n = [E:F]$ over $F.$ Let $C$ be the companion matrix for $m(x)$ which is an $n \times n$ matrix with entries in $F.$ Then whenever $\alpha^{j}$ appears as an entry of a matrix in the given representation over $E,$ replace it by the $n \times n$ block $C^{j}$ (and put an $n \times n$ block of zeros whnever $0$ occurs in the original matrix representation). This gives the representation over $F$ of dimension $[E:F]$ times the dimension of the original representation, which is unique up to equivalence.

One uses a matrix version of Hilbert's Satz 90 to answer Geoff's comment "If the field $E$ is not this minimal field, it seems less obvious to me how to realise the representation over the subfield generated by the traces, and I suppose this may be the real issue you need to resolve." An easy-to-read description is given in S.P. Glasby and R.B. Howlett. Writing representations over minimal fields, Comm. Algebra 25 (1997), 1703--1712.