The command \definecolor{shadecolor}{rgb}{0, 0, 1}
issued in a TeX inset (Ctrl+L) before the shaded box will change this color to blue. If the change should be only local for that particular box, do Ctrl+L; {
; exit the TeX inset; Ctrl+L again; \definecolor{shadecolor}{rgb}{0, 0, 1}
; exit the TeX inset; insert your box; Ctrl+L; }
I do not understand why LyX puts the environment \begin{shaded} \end{shaded}
of framed.sty inside of a minipage, as the environment was meant to spill freely across the pages.
You could just as easy insert another kind of box, say a frameless one, and change its background color with a right click.
EDIT: OK, you can choose the 'inner box' to be None
, and the environment will go across the pages.
Pictures added to illustrate:
In LyX

PDF output of above LyX file:
