They start from the simplest, and get more complex - actually that depends on your point of view - and I'm talking in Moodle terminology here, but if you're using a different tool then you'll be able to map this onto your system, won't you? Good!
- Set up offline assignments to make a comprehensive homework diary
The simplest form of Assignment in Moodle is the Offline Assignment - there is little or no interactivity in terms of handing in the work, but as a teacher you can use the Description field of the assignment to give useful links to start from, to embed videos to watch, or audio clips to listen to, and provide all of the guidance you'd normally give when handing out a paper-based piece of homework in the classroom. Importantly, set a due date so that the homework will appear in the course's Calendar and Upcoming Events - this way, a student (or their parents if your setup includes parental access) can see what's coming up over the next month / term / year (depending on how far you plan in advance). As the assignment behaves like a normal Moodle assignment, you'll be able to give marks and feedback, which can be stored in the gradebook so that students (and parents... etc) can see all of their feedback & marks in one place - plus as a teacher you get an overview of all homework in the Gradebook.
- Set up a Quiz with opening & closing times as a homework
When setting up a Quiz it's a trivial thing to to set opening and closing times - set these to Friday 5pm and Monday 8am and you've a piece of work which, by definition, has to be done over the weekend. You need to be sure that all of your class has internet access outside of school for this to be valid - if this isn't the case, make arrangements with a local library or explore how students could buddy up outside of school to deal with this. This requires more preparation (of the questions, for a start...) but has the advantage that it's marked automatically (if the question types suit automatic marking). Again, all of the marks end up in your Gradebook (but can be modified if you feel that some attempts merit this...).
- Assess contributions to a Forum
The same "access to online" proviso as #2 applies here, but setting up a forum as "homework" has a bit more bite if you mark responses using the Rating tools built-in (but rarely used) into the Forum module. Such Ratings are recorded in the Gradebook and by default only teachers can Rate/grade forum contributions. It's always a good idea to model appropriate and inappropriate responses first - so that you've got exemplar material to mark against. Ratings can be available as long as the forum's open, or restricted to a particular date range, this might be useful as you are likely to have to work with forums for a while before students get used to the idea that a meaningful response to another student's message is more than just "Yer Mum". It might be worth having a few practice forums so you can effectively manage students (and your) expectations before having an assessed exercise.
- Build a glossary together as a class
The Glossary activity is an under-used activity in Moodle and allows students to build a user-defined dictionary of... well, anything really. You could ask students to contribute to a collaborative glossary of biographical entries for a list of characters in a play, relevant information about places in a series of historical events, or anything really. The key with this sort of exercise is to establish at the start that simply copying material from Wikipedia or Encarta isn't the way to go - so try putting a spin on it by limiting it to a number of words (like Four Word Film Reviews, how about Thirty Word Biographies?). The power of the Glossary is that it can be used throughout your online course, so any time a term in the Glossary occurs - in a Forum post, say - then a link is made to that term's definition which everyone can read. Entries can be graded, so you can set contributing to a glossary as a piece of homework.
- Hand in homework online
This is the sort of activity which most people think of when they think of "Homework on a VLE" - setting an Assignment with a due date and instructions, pretty much as #1 above, then collecting the work on the VLE. In Moodle this is done via a simple Online Text assignment, an Upload a Single File assignment, or an Advanced Uploading of Files assignment. The details of these are best explored by reading the documentation pages, or simply by setting up an assignment of the kind you're interested in, but there are a number of creative practices which can be done with these tools. The Online Text assignment is fairly simple, but as it uses the HTML editor for students to enter their work in, there are lots of creative possibilities. If you're using Google Docs with your students, you could set an assignment (create a spreadsheet / presentation / etc.) as an online text Assignment and then students would respond with a link to their presentation and then a paragraph of explanatory text. You could set an audio homework by using something like GabCast - students simply link to the contribution that they made via their phone or (for the geeks among them) embed it in the HTML editor (if your Moodle site is set up to allow embedding to be done by students). There are as many ways to set up creative pieces of homework as there are creative tools on the web - however it's a good idea to settle on a few and stick to them, rather than aiming for something novel every week. That way students have more chance of focusing on what you want them to study, rather than the tool you want them to learn.
One word of advice would be - don't try everything at once. Think about what you would do with "normal" paper-based homework, work out which tool(s) would work and then have a go - starting simply and developing your use and you (and your students) learn how things work.
What other ways might you use a VLE for homework? Has anything worked better than you thought, while other things died a slow and painful death?