Consider

'ab' =~ /a(z?)b/   # Returns ''.
'ab' =~ /a(z)?b/   # Returns undef.
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So (\([^\n]+\)) is optional? Then let's add a ? as follows:

my @fizzbin = ( $string =~ m{
    (\w+)\n                     # Grab the title
    What:\s+([^\n]+)\n          # Grab the what
    Date\sadded:\s+([^\(^\n]+)  # Date added
    (\([^\n]+\))?\n             # Optional source
    Data:\s+([^\n]+)\n          # Grab the data
}xmgs);
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After looking at my first snippet, we determine that the source will be undef if the source is omitted. Let's rearrange your code to use named variables like you wanted, and let's integrate the default. We get the following:

while (
   my ($title, $what, $date, $source, $data) = $string =~ m{
      (\w+)\n                     # Grab the title
      What:\s+([^\n]+)\n          # Grab the what
      Date\sadded:\s+([^\(^\n]+)  # Date added
      (\([^\n]+\))?\n             # Optional source
      Data:\s+([^\n]+)\n          # Grab the data
   }xmgs
) {
   $source = '' if not defined $source;  # Use default ''.

   ...
}
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By the way,

• the m modifier is useless, since you don't use ^ or $;
• the s modifier is also useless, since you don't use .; and
• Date\sadded could be replaced with Date[ ]added to improve readability.