What are some free (or cheap) options for hosting home videos that have copyright music in them?
You can use a file storage service like:


DropBox 
Google Drive 
Microsoft SkyDrive 
Apple iCloud


You can restrict access of your files there to a handful of persons.

For example on Google Drive, when you share a file with some people, it asks you for their email addresses. It then sends an email to all those people with a "secret" link. In theory, this link is still public. But in practice, it is next to impossible to guess the value of those links.

Personally, I use Google Drive. It offers an initial 15 Gb of space.