ScriptMinSize is an option for StyleBox which specifies the minimum font size to use in rendering subscripts, etc.
Settings for ScriptMinSize are in units of printer's points.
ScriptMinSize is used for characters that appear in constructs such as subscripts, superscripts, underscripts, overscripts and built-up fractions.
ScriptMinSize is typically set larger in styles used for screen display than in those used for printing.
See The Mathematica Book: Section 2.8.11.
See also: ScriptSizeMultipliers, FontSize, ScriptBaselineShifts.