Write lyrics first. Since all of the melody and harmony for the song will be in the singing of the words, this is an important element. Choose a subject for the song you want to write about. Many a cappellas are spiritual in content, but any song sung without music is a cappella, so you aren't limited in subject matter. Brainstorm for lyric ideas and write down three or four lines that fit your song subject..
Structure the lyrics like a poem. A lyric for an a cappella might look like this:
The time has come to say farewell
to all the memories of the past.
I've got to leave this broken town.
But my car is out of gas.
You can rhyme or not. Like any song, rhymes are not necessary, but they can make the song more singable.
Sing what you have as you write it. Snap your fingers in time as you sing to help develop the melodic and rhythmic structure of the verse. You may find the lines you wrote have words with too many syllables to fit the melody want. Change those words that don't fit within the melodic structure or change the rhythm of the melody to fit the lyrics. Writing is a process, so you'll find yourself changing the melody or lyrics frequently. That's OK. It's part of perfecting your a cappella.
Record your a cappella as you complete verses. This is the best way to ensure you don't lose the melody. Play back what you've written and sing along. Find ways to enhance the main vocal by adding background vocals. The background might be as simple as singing some of the same parts in a higher or lower register. Writing more than one vocal part, even if it is simple lyrics like "shoo-wop shoo-wop" sung beneath the chorus, can help add some musical quality to your song.
Sing your finished a cappella with a couple of friends. Assign a part of each friend, rehearse the song and record it to hear what it sounds like put together. If your friends are interested, ask them to write with you. Work as a team to make changes or to perfect the parts you have written. Try different techniques, such as having the bass singer do a counter melody to add a sense of movement. Some of this can be improvised once you have the main melody and harmony in place.