Survey the competition. Go to the local bookstore and peruse the children's section for books that are similar in genre and theme to the book your child has written. Make note of the publishers of these books.
Subscribe to monthly publications such as Children's Writer Newsletter, which provide editor wish lists, publisher interviews and submission guidelines. This entity also publishes annual resource guides for books and magazines.
Bookmark websites such as Teens Read, Too! (see Resources). In addition to its companion link to Kids Read, Too! you'll find book reviews, author interviews and writing contests that will supplement you and your child's education insofar as what's being published (and being successful).
Purchase a copy of Children's Writer's and Illustrator's Market by Alice Pope. This is published annually by Writers Digest Books and will supply you with comprehensive submission information for age-appropriate markets that your child's book would be a good match for.
Solicit honest feedback from your child's friends and your peers on whether this book has potential for publication. As a parent, you're probably a tad biased on the quality of the material and it's important to find out what others think before you decide to try the traditional publishing route or self-publish it.
Research each of the publishers you think would be a prospective buyer for the book. Study their submission guidelines carefully and follow them to the letter. If they say they only want to see a cover letter and the first chapter, you're not going to score any points by sending them the entire manuscript. Likewise, if they state that they only read material that has been submitted to them via an agent, do not send it in yourself and think they're going to make an exception for you.
Write a short, professional cover letter that provides a short synopsis of what the book is about, how long it is, whether there are illustrations and a brief background on the author. You will need a cover letter whether you are submitting directly to an editor or to an agent.
Research agents who represent children's books. Websites such as Agent Query are a great place to start. In the left-hand menu on the opening page, click on Fiction and you'll find Children's genre. Click on this and it will generate a list of agents currently seeking children's literature submissions.
Print out fresh copies of the manuscript each time someone asks you to submit material.
Include a self-addressed, stamped envelope with each letter or copy of the manuscript you send off. While there are many publishing houses that now accept electronic submissions to save time, traditionalists still prefer snail mail copies. They will state their preferences in their submission guidelines or on their websites.
Research self-publishing entities if (1) the book isn't unique or commercial enough to get picked up by a traditional publisher, or (2) you have enough family, friend and coworker connections to sell copies so that you can recoup your investment. Take heed that not all self-publishers are equal. Some of them charge exorbitant set-up fees, require you to purchase your own ISBN numbers and expect you to buy and warehouse "X" number of copies yourself. Others, such as Outskirts Press, offer a multiple range of publishing packages, allow you to control your own pricing and royalty structure, provide ISBNs and make books available through major distributors and have access to editorial tips and advice to help you promote your child's project.
Research ebook companies. Like self-publishing entities, there's a wide range of prices and services out there. Unlike traditional houses, few ebook companies pay an advance to authors but all of them offer a quarterly system of royalties based on the number of ebooks that are either uploaded at purchase or sold as CDs.
Consider "publishing" the book yourself by either having copies printed at a neighborhood photocopy shop or turning the material into a CD-ROM. These can then be sold off your own website or at community fairs. The advantage of CD-ROMs is that they are inexpensive to burn copies and you can generate them on an as-needed basis instead of having to find a spare room to store them until they sell.
Consider "publishing" the book through creative entities such as My Publisher. This is a service that allows you to upload text and photos, pay a fee based on the length and format of the book and then receive a professionally bound copy within a week. While it could get cost-prohibitive if you planned to order a lot of copies, the thrill of seeing a "real" book sitting on the coffee table may be all your eager young author really needs at the moment to feel happy and confident.