The smallest integer that does not become prime when changing a digit is 200.

Mathematica code

test[n_] :=
   Union[PrimeQ[FromDigits /@
        Flatten[(
                Join[Take[IntegerDigits[
        n], {1, # - 1}], {
          a}, Take[IntegerDigits[n], {# + 1,
                Length[IntegerDigits[n]]}]] & /@ \
Range[Length[IntegerDigits[n]]]
                  /. a -> #) & /@ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, 1]
      ]]

tbl = Table[{n, test[n]}, {n, 1, 1000}];

Select[tbl, Length[#[[2]]] == 1 &]

-- 
Chris Lomont
http://www.lomont.org