Although I agree that probably a regexp is more straight-forward, I get different results benchmarking the stuff.

my $string = q{"string (12" or 1 foot or 1') into 6" MySQL varchar"};

benchmark(
        shift,
        q{"string (12"" or 1 foot or 1') into 6"" MySQL varchar"},

        'substr' => sub {
                my $t = $string;
                substr($t, 1, -1) =~ s/"/""/g;
                $t
        },

        's///' => sub {
                my $t = $string;
                $t =~ s/(?<=.)"(?=.)/""/g;
                $t
        },

        'index' => sub {
                my $t = $string;
                local $_ = 1;
                while( ($_ = index($t,'"',$_)) !=-1
                               and ++$_ < length($t)
                                      ){
                              substr($t,$_++,0,'"');
                }
                $t
        },
);
[download]

benchmark() is just a wrapper around Benchmark::cmpthese() that checks the results of the routines.

# run with perl 5.6.1

EXPECTED   : "string (12"" or 1 foot or 1') into 6"" MySQL varchar"
index      : "string (12"" or 1 foot or 1') into 6"" MySQL varchar"
s///       : "string (12"" or 1 foot or 1') into 6"" MySQL varchar"
substr     : "string (12"" or 1 foot or 1') into 6"" MySQL varchar"

Benchmark: timing 1000000 iterations of index      , s///       , subs
+tr     ...
index      : 19 wallclock secs (19.66 usr +  0.02 sys = 19.68 CPU) @ 5
+0813.01/s (n=1000000)
s///       : 19 wallclock secs (19.85 usr +  0.00 sys = 19.85 CPU) @ 5
+0377.83/s (n=1000000)
substr     : 23 wallclock secs (24.13 usr +  0.00 sys = 24.13 CPU) @ 4
+1442.19/s (n=1000000)
               Rate substr      s///        index
substr      41442/s          --        -18%        -18%
s///        50378/s         22%          --         -1%
index       50813/s         23%          1%          --

# and with perl 5.8.0

EXPECTED   : "string (12"" or 1 foot or 1') into 6"" MySQL varchar"
substr     : "string (12"" or 1 foot or 1') into 6"" MySQL varchar"
index      : "string (12"" or 1 foot or 1') into 6"" MySQL varchar"
s///       : "string (12"" or 1 foot or 1') into 6"" MySQL varchar"

Benchmark: timing 1000000 iterations of index      , s///       , subs
+tr     ...
index      : 18 wallclock secs (18.71 usr +  0.01 sys = 18.72 CPU) @ 5
+3418.80/s (n=1000000)
s///       : 33 wallclock secs (31.67 usr +  0.02 sys = 31.69 CPU) @ 3
+1555.70/s (n=1000000)
substr     : 35 wallclock secs (35.24 usr +  0.01 sys = 35.25 CPU) @ 2
+8368.79/s (n=1000000)
               Rate substr      s///        index
substr      28369/s          --        -10%        -47%
s///        31556/s         11%          --        -41%
index       53419/s         88%         69%          --
[download]

Okay, I re-ran the benchmark with your exact code on my computer, and I got:
Read more... (1391 Bytes)