Please let me know if I'm the right track here, or how it should be done better.

#!/bin/bash
list=`ls -al /some/directory | grep "$1"* | awk '{print $9}'`

for file in $list
do
mail -s "$file" me@email.com < $file
done

file1
file2
file3

Earnstaf,
To answer your question, yes, it does.
Keep in mind that the best way to learn is testing it.
Run the following:

list=`ls -al /some/directory | grep "$1"* | awk '{print $9}'`
for file in $list
do
  echo "Working with file = "$file
  #####mail -s "$file" me@email.com < $file
done

Thanks Shell Life. I'm getting some unwanted results. I've modified the script to look like this:

#!/bin/bash
list=`ls /some/directory | grep "$1"*`
echo $list
echo "starting loop"
for file in $list
do
echo "Mailing" $file
#mail -s .......
done

./mail.sh file1

starting loop

I understand your problem now.
Here is the solution:

#!/bin/bash
list=`ls /some/directory/$1*`
echo $list
echo "starting loop"
for file in $list
do
  echo "Mailing" $file
  #mail -s .......
done

Very good, it works that way Shell Life.

Is it not possible to pipe within the variable expansion?

Edit: or if I have to make it more specific, like if I need to do something like "grep "$1.*-ingress-filter" to restrict the output to the list variable.
Edit 2:
I've tried the following but it doesnt seem to do the trick.
I set list as you specified above, then this:

list2=`echo $base | grep "$1.*-filter"`

list=`ls /some/directory/$1*-filter`

Earnstaf,
You are confusing file name with file content.
The 'grep' command reads file contents.
The 'ls' command reads file names.

I think that the problem comes from the pattern * of your grep command.
The shell expand "$1"* to files starting with the value of $1 in your current directory.
Remove the * :

list=`ls /some/directory | grep "$1"`