system "pdftk /home/Memos/exhibit1_29JUN2010.pdf /home/Memos/exhibit2.pdf cat output /home/exhibits_29JUN2010.pdf";

I've tried using a wildcard (*) to take the place of the datestamp (29JUN2010) in the first file's name, but I'm still having the problem of getting the combined file to have the datestamp in its name.

Not sure I understand. Have you hardcoded those file names in the system call?  I would guess that you get the input file names from somewhere (listing directory contents, some config file, user input...).  Otherwise, how would the program work if there's a new date stamp...

So why not simply dynamically create the output file name according to the date stamp found in the input file?  For example:

my $in1 = "exhibit1_29JUN2010.pdf";
my $in2 = "exhibit2.pdf";

my ($date) = $in1 =~ /_([^_]+)\.pdf$/;  # extract date

my $out = "exhibits_$date.pdf";         # create output file name

system "pdftk /home/Memos/$in1 /home/Memos/$in2 cat output /home/Memos
+/$out";
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Or maybe

my $dir = "/home/Memos";
my $in1 = <$dir/exhibit1_*.pdf>;
my $in2 = "$dir/exhibit2.pdf";

my ($date) = $in1 =~ /_([^_]+)\.pdf$/;

my $out = "$dir/exhibits_$date.pdf";

system "pdftk", $in1, $in2, "cat", "output", $out;
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