The least expensive option is self-publishing using a free service like Amazon KDP, coupled with cheap illustrations. It will cost you zero dollars in terms of publisher's fees, although you should consider spending some money on editing and cover creation because you don't want your book to look like an amateur effort after all is said and done.

Since illustrations will likely make up the bulk of your costs, this is the only thing you should spend time thinking about when working on a tight budget. Some writers try to find a partner who can split royalties with them on their children's book instead of paying upfront for their services or even hire someone to illustrate a sample first and then commit to publishing the whole book. When working with children's book publishers online, you should always compare the price packages from several children's book publishing services (those that offer book illustration packages in particular).