Hi! Let me start off by saying that I have a pirate theme in my classroom so that's why so many of the activities sound pirate-y
I send our class mascot (a stuffed animal named Pete the Parrot) home in a bag with a composition notebook and the students get to write about what Pete did with them that evening. I don't send him home over the weekends because we don't have homework on weekends. Each morning the person who took pete home reads what Pete did with them the night before. This is a fun activity that the children enjoy.
I have an activity called "treasure hunt" where I ask students to take their homework notebooks and search for things in their homes that fall into a certain category and make lists. They might have to search for things that are magnetic or words that start with the letters in their first name. I've made them find words that start with ch/sh/th and wh. They've searched for needs and wants, words with different numbers of syllables-- basically anything in the curriculum that can be tied into their home. After they make a list I let them choose 3 to illustrate for me.
Our school also requires a lot of projects from our students so every 2-3 weeks instead of homework, my students have a week to complete a poster or project of some type that supports the curriculum. Right now they are designing their own poll and making a graph poster to display the results of their poll. It never ceases to amaze me what soem of my children are capable of without the help of their parents! (it's usually easy to tell which parents have "helped" their children)