BUT ... that means my students go home with an address for their blog that looks like myblogname.blogspot.com - in other words, they will have the "blogspot" bit in their domain name, rather than having a custom domain (which would be myblogname.com).
This is not that cool. It's similar if you use the free wordpress version, and you have seen people with blog URLs like myblogname.wordpress.com. Whichever you use, I can't recommend enough that you should get yourself a custom domain. For lots of reasons. Most of all, you look a thousand times more professional and your blog's URL is a million times easier to remember and tell people. (Okay, these numbers may be slightly exaggerated, but my point is that I consider this really important!) The main reason I recommend getting your own domain name is that these days it is so simple and cheap so there's no excuse not to - and it's pretty common that what's started as your little hobby blog becomes something much more important to you. You can get sites redirected and so on but it's just simpler to start how you mean to go on - and if you can afford the time to blog, you can afford the minimal cost of this, I reckon.
The lucky thing is that if you are using Blogger then it's super simple and not even expensive (they currently charge US$10 per year for both the custom domain name plus they host it for you - there is no better deal than that). The "how to" goes like this:
1. In Blogger, choose "Settings" (and the first sub-menu "Basic" will appear by default)
2. On the "Basic" page of settings, look for "Publishing" - there will be a section for "Blog Address"
(PS: Experienced bloggers will immediately tell you that you "must" have Wordpress rather than Blogger, for two main reasons - a Wordpress blog can be basically endlessly customised and also you have more control, because you have to host it somewhere of your choice rather than letting Blogger (aka Google) host it for you. These two points are both absolutely true, but in the case of most of the people I work with, they are not in the mood for learning the more technical aspects of Wordpress, and they are often working from lower budgets. In other words, for many of them, a custom domain on Blogger is enough - and in any case, it can all be transferred later if they so desire.)