I'd partition the data into a number of buckets that match the number of points you want to end up with, then select the best value from each bucket according to whatever weighting is appropriate. Consider:

#!/usr/bin/perl
use strict;
use warnings;

my $numBuckets = 20;
my @points =
    map {{x => $_->[0], 'y' => $_->[1], dy => $_->[2]}}
    map {chomp; [split]} <DATA>;
my @buckets;
my $min = $points[0]{x};
my $max = $points[0]{x};

for my $point (@points) {
    $min = $point->{x} if $min > $point->{x};
    $max = $point->{x} if $max < $point->{x};
}

my $scale = ($max - $min) / $numBuckets;

push @{$buckets[($_->{x} - $min) / $scale]}, $_ for @points;

for my $bucket (@buckets) {
    # Sort contents of bucket by weighting function
    next if !defined $bucket;
    @$bucket = sort {$a->{dy} <=> $b->{dy}} @$bucket;
}

for my $index (0 .. $numBuckets - 1) {
    printf "%3d: ", $index;
    printf "%.4f, %.4f, %.4f", @{$buckets[$index][0]}{qw(x y dy)}
        if defined $buckets[$index];
    print "\n";
}
[download]

using the data in the OP prints:

  0: 0.0345, 0.9916, 0.0013
  1: 0.0499, 0.9876, 0.0011
  2: 0.1340, 0.9659, 0.0012
  3: 0.1635, 0.9578, 0.0011
  4: 0.2149, 0.9412, 0.0047
  5: 0.2911, 0.9215, 0.0015
  6: 0.2974, 0.9186, 0.0010
  7: 0.3617, 0.8983, 0.0018
  8: 0.4183, 0.8819, 0.0010
  9: 0.4535, 0.8672, 0.0085
 10: 0.5317, 0.8421, 0.0010
 11: 0.5689, 0.8306, 0.0040
 12: 0.5995, 0.8179, 0.0056
 13: 
 14: 
 15: 
 16: 0.8015, 0.7142, 0.0249
 17: 0.8540, 0.6901, 0.0060
 18: 0.9126, 0.6475, 0.0020
 19: 0.9690, 0.5879, 0.0023
[download]

which has drawn too few points because the actual distribution in the original data is very lumpy. If you need a fixed number of points and it is likely that you won't get at least one datum in each bucket, then I'd select further points from the buckets with the greatest number of points in them.