The lock box analogy
One of the most powerful ways to leave an impression on children is to show them, through demonstration, that an idea which they likely hold as self-evident common sense is completely false. In this case, the idea that all bets are off if you are passing a secret message to someone via a curious or malicious messenger.
I find that many kids and people not generally security-savvy tend to have the hardest problem understanding asymmetric cryptography. It's easy enough to trust that a password can be used to scramble data, but idea that two people can exchange a shared secret in a hostile communication medium is not at all intuitive. An analogy I have used successfully exploits the one-way behavior of common locks (i.e. anyone can lock it, but only those with the corresponding key can unlock it).
A hands-on demonstration
It's useful here to describe how a shared secret can be communicated, using a hands-on approach. I wouldn't make it particular to TLS (as the handshake involves a lot more), but rather general among all asymmetric ciphers. You play Alice, one kid plays Bob, and the other plays Eve. Tell the kid playing Bob that his goal is to communicate with you in secrecy without letting Eve know what you are saying, despite Eve being the one passing the messages along. The demonstration is simple:
Give an unlocked lock box (but not the key!) to Eve and tell her to pass it to Bob.
Have Bob write down a secret note, put it in the lock box, and close the lock.
Have Bob give the lock box back to Eve and ask her to pass it back to you (Alice).
Use the key to unlock the lock box and look at the note.
Encrypt a message with a simple substitution cipher, using the note as the key.
Give the encrypted message to Eve and ask her to pass it to Bob.
Let Bob decrypt the message. Talk back and forth a few times using the shared secret.
You can skip the substitution cipher part if you're short on time, and instead focus on the fact that you managed to share a secret without Eve being able to figure out what it is, but explaining the basics of symmetric cryptography may still help kids understand the theory behind it better than just telling them to trust that a short secret value can be used to reversibly scramble a larger message.
If you're lucky, a clever kid might ask why Eve can't just swap the unlocked box with one she owns, in which case you get a useful opportunity to explain MITM attacks and digital signatures. The solution becomes easier to understand when they have a real-life analogy of a trapdoor function: something that is easy to verify, but hard to forge. A written signature fits this perfectly. I don't think there's a need to explain certificates or CAs. Just make it general to all digital signatures, so their understanding will apply equally to TLS, signed executables, or GnuPG.
Perhaps also mention that browsers will give you an indication when this process fails, since a misunderstanding of that is commonly exploited in phishing attempts.
What Eve represents
You may have to explain that connecting to a website requires servers in between to pass the data back and forth. While it may be obvious to us, a 10 year old may think that your computer communicating with any website naturally involves a direct connection, with no entity in between. Explaining that there are computers you may not trust in between you and the destination will let them better understand what Eve represents.
You can expand on this a bit by mentioning that the one passing the messages back and forth isn't necessarily malicious. Eve can very well just be someone snooping on the messenger (sniffing Wi-Fi, or using another person's router, for example).
While it may be out of scope, you could show how such simple substitution ciphers can be broken, and then give them something like sending you an encrypted Solitaire message as homework (or suggesting that to the teacher, if you don't have the authority to distribute homework), so they can see what a cryptographically strong, albeit toy, cipher is like. It can leave a strong impression on a child to know that no one knows how to break a message that they encrypted with their own hands! A sense of power while learning is very appealing to children.
Bonus points for using the names Alice, Bob, and Eve in your discussions. ;)