There are many ways to make a logo for cheap. Branding is a big business with graphic design companies charging up to millions of dollars for one logo in some cases. However this is not the type of logo you require. You are after a simple professional logo that creates a brand for your work.
I would not recommend you try to create your own logo using MSword or MSpaint. Although this is possible it takes away the point in creating a logo in the first place. You want it to look professional not amateur. These tools are not ideal for making a logo. If you are familiar with Adobe Illustrator or Photoshop then you can defiantly make your own logo as these programs are very well suited for logo design. If you are unsure of how to use these programs to make a logo check out Lynda.com. They have great tutorials on how to make a logo.
There are many good logo providers online who can create professional logos for a very reasonable price. They do all the hard work for you and you don't need any technical knowledge at all. A lot of them have a wide range of options for both the text and graphic. One I like to use is ‘Logo Maker’ Professional Logo in Minutes!
Things to think about when designing a logo
What does your publishing company represent?
Who is your target market? (Usually the buyer not the reader in this case)
What colour would fit with your books look and feel?
Is there a graphic object or element that could represent your company in a simple sophisticated way?
Tips and Tricks
Try to keep your logo to one to two colours. This will keep printing costs down later down the track when you require stationary and other promotional products. It also makes it easy to place on colourful children's books without having any colour clashes or higherachy problems.
Keep the name of your company short and simple. You want to be able to place your logo on books and products at a very small size. Too much text can make this difficult.
Keep any images and graphics simple for the same reason as the last point. You want this logo to work at a very small size.
Try and make your logo fit into a square or rectangle as best as you can. This is helpful when trying to place it on a page or spine. The bigger the text fits in the space the more legible it will be at a small scale (on a spine).