You're pretty close,

for (@lines) {
    s/(\d+\s+)(.+)$/$1(Birthday) $2 - singer/;
    print;
}
[download]

The following seems to be what you are after:

use strict;
use warnings;

my @lines = <DATA>;

foreach my $line (@lines)
{
  next if $line !~ m/(\d+\s)\s*(.+?)\s*$/;
  print "$1(Birthday) $2 - singer\n";
}

__DATA__
January 1, 1915      Franck Pourcel
January 3, 1926     George Martin
January 3, 1945     Stephen Stills
January 3, 1946     John Paul Jones
January 4, 1942     John McLaughlin
January 5, 1950     Chris Stein
[download]

Prints:

January 1, 1915 (Birthday) Franck Pourcel - singer
January 3, 1926 (Birthday) George Martin - singer
January 3, 1945 (Birthday) Stephen Stills - singer
January 3, 1946 (Birthday) John Paul Jones - singer
January 4, 1942 (Birthday) John McLaughlin - singer
January 5, 1950 (Birthday) Chris Stein - singer
[download]

However you probably want to read up a little about sprintf (to clean up the formatting). You might also want to read the Tutorials sections here for an introduction to regexen. Also read the Perl man pages perlretut, perlrequick and perlre.

Presuming your code was copied correctly, it's not syntactically correct: you're missing the closing slash after the $1. When that's corrected, if I read it correctly (a very big if!), you'll just replace the year and everything to the right of it by the year, and the trailing blanks.

Try this:

#!/usr/bin/perl

use strict;
use warnings;

while(<DATA>){
    chomp();
    print "before: $_\n";
    s/(\d+\s+)(.+)/$1 (Birthday) $2 - singer/;
    print "after: $_\n";
  }

__DATA__
January 1, 1915      Franck Pourcel
January 3, 1926     George Martin
January 3, 1945     Stephen Stills
January 3, 1946     John Paul Jones
January 4, 1942     John McLaughlin
January 5, 1950     Chris Stein
[download]

I think this will make the change you want; the regex is the important bit.

#!/usr/bin/perl

use warnings;
use strict;

open(LOG, "source.txt") or die "Error: $!";
my @lines = <LOG>;
close(LOG);

my $cnt = 0;
foreach my $line (@lines)
{
  $cnt++;
  $line =~ m/(\d+\s+)(.+)/;

  $lines[$cnt] = "$1 (Birthday) $2 - singer";

  
}

open(LOG, "> source2.txt") or die "Error $!";
print LOG join("\n", @lines);
close(LOG);
[download]

# in the cycle:
...
 s/(\w+\s+\w+\W*).$/(Birthday) $1 - singer/
...
[download]