As an option
Interpretation[{x = RandomReal[]}, Button[ToString@SetPrecision[x, 6], x = RandomReal[]], N[x, 6]]
Or
Interpretation[{x = RandomReal[]}, Button[N@Round[x, 1*^-6], x = RandomReal[]], N@Round[x, 1*^-6]]
As an option
Interpretation[{x = RandomReal[]}, Button[ToString@SetPrecision[x, 6], x = RandomReal[]], N[x, 6]]
Or
Interpretation[{x = RandomReal[]}, Button[N@Round[x, 1*^-6], x = RandomReal[]], N@Round[x, 1*^-6]]