I'm teaching some introduction lessons to a classroom of 6-8 year olds about basic 21st century skills. A fundamental part of this is how computers represent numbers, words and images, which all starts with understanding binary representation of a number.
So I'm thinking of using different symbols. Like ● for 'this bit is on', and ⊗ for 'this bit is off' ...
My son just turned 8 and I've run into this same problem many times. I've found it an interesting and rewarding challenge to try to explain concepts without getting too deep into technical details.
For my son, (and even many adults) I've found it extremely valuable to not talk much (if at all) about binary, place value, and number bases (at least at first) as bytes are what everyone uses and talks about near daily. I've also found it good to avoid talking about specific numerical quantities and focusing instead on what those quantities represent, such as letters, brightnesses, etc. In some conversations I also found it useful to explain that bytes in a computer are taking up physical space inside the memory of the machine, just as letters on paper take up space.
Teaching Binary Numbers:
Sometimes light bulb on/off symbols are used ...
An example from Basic Computing book is coming to my mind here: Suppose you have some torches or light bulbs and want to convey a message using the combination of those bulbs/torches/LEDs.
-- Failed Scientist
... I have never not confused anyone with notation.
When I tried to teach my son binary in order to explain how numbers worked in a computer I found that what worked best was to keep things simple and only build from ideas he already knew and nothing else. The usual light switch (or LED) analogy seemed to help some, but working with our hands was best. I considered using blocks, drawing pictures, and so on, but ended up not needing to.
Simple counting, doubling, and adding were things he already knew. Place value, not so much, but it wasn't hard for my son to get the general idea that each of his fingers could represent a number twice as much as the one before it, and you add them all together after you hold them up. Each finger can either be up or down; added or not.
He was then very impressed with how we could count to 31 on just one hand, and 1023 using both hands. This understanding actually came after explaining what bytes were and getting him to understand them.
Teaching How Computers Store Letters, Words, and Images
The really important part was to explain that a "byte" was a number in the computer, and that it could represent different things, like one letter, or how bright a color was. Explaining that a byte was 8 bits came later when he began learning what bits were.
As for words, I told him that you could lay out the letters of the alphabet in a row and count them, and that's how computers know what letters are, because the people who build computers decided which numbers would be which letters. I was then able to explain from there how many bytes it would take to make various kinds of documents around our house.
Explaining pictures also started with a recap on bytes. I explained that 1 byte told the computer how much of a color was in a picture. The computer would read that number and see how much red was in the picture. I next Googled "color mix venn diagram" so we could have a conversation about how pictures are only made of red, green, and blue, and thus 3 bytes are needed to talk about color. From there I was able to explain that those 3 bytes don't actually describe the entire picture, just one very tiny dot, and that you then need a lot of bytes to draw one image.