Some thoughts:

• Is the Son => Father key/value relationship that I see as the basic organization of your data structure valid? My first thought would have been Father => Son: A father may have many sons, but a son has only one father (if we ignore step-fathers, fathers-in-law, (update: godfathers,) and similar relationships from other cultures of which I am unaware — and always remembering that it's a wise child that knows its own father).
• You do not  use warnings; Had you done so, Perl would have warned you that the statement
      $son_father{$name1} = {$add_dad};
  isn't quite kosher:

  c:\@Work\Perl\monks>perl -wMstrict -le
  "use strict;
   use warnings;
   ;;
   use Data::Dump qw(dd);
   ;;
   my %son_father;
   ;;
   my $name1   = 'Jeff';
   my $add_dad = 'Doug';
   ;;
   $son_father{$name1} = {$add_dad};
   ;;
   dd \%son_father;
  "
  Odd number of elements in anonymous hash at -e line 1.
  { Jeff => { Doug => undef } }
  [download]

  If you're learning Perl, always  use strict; and  use warnings;
  If you're not learning Perl, always  use strict; and  use warnings;
• Spurious indentation may be leading you astray. Consider the following re-formatted code:

  while (1) {
      # display the menu
      print $menu, "\n\n";
  
      # get the user choice 
      print "Make your choice: ";
      chomp($choice = lc <STDIN>);
  
      # fulfill the user request 
      if ($choice eq 'a'){
          print "Enter a male name: ";
          chomp ($name1 = lc <STDIN>);
          }
  
      if (exists $son_father{$name1}) {
          print "Duplicate name -- try again!\n";
          } 
      else {
          print "Add a father: ";
          chomp ($add_dad = lc <STDIN>);
          $son_father{$name1} = {$add_dad}; 
          } 
          
      ...
      
      }  # end while loop
  [download]

  Is it more clear that after the  $choice eq 'a' if-test is done, the  exists $son_father{$name1} if-test will always be done and, at least the first time through the loop, will always be false, so the else-clause  print "Add a father: "; will always execute? In a similar way, work through the other conditional statements in the loop body.
• Update: Perhaps a few print-points for debugging would be helpful, or  print Dumper ...; (see Data::Dumper), or  dd ...; (see Data::Dump)?

All I can think of for the moment. HTH.