I want to create a custom field to be indexed in Lucene that has the children of the item being indexed. Is there an easy way to do this that I am overlooking?
According to your requirement, you should look into creating a custom computed index field.
This community article by John West describes the process of adding a custom computed field in details: https://community.sitecore.net/technical_blogs/b/sitecorejohn_blog/posts/sitecore-7-computed-index-fields
Basically, all you have to do is to define a computed index field, with your custom logic in the
ComputeFieldValue method (in your case -> iterate through item's children and collect all you need, e.g. names, etc.)
The above article gives an example for Sitecore 7+, but in Sitecore 8+ the process is not very different