Memorising a really strong master password is probably a bit much to ask
I disagree! I have a daughter who, at around 7, was able to quickly memorize a very strong password using the Diceware method for use in a password manager. This method works by picking several random words from a dictionary typically composed of 7,776 words. This is also, by no coincidence, the number of possible results of five independent rolls of a 6-sided dice. As such, you can use real dice to generate your passwords (and although you can buy casino-grade die if you wish, the bias is so small that it doesn't really matter). A mere 9 words (45 dice rolls assuming you have just one dice) provides log2(77769) ≈ 116 bits of security which is more than adequate for a password.
If you use a password manager that supports password-strengthening with a slow KDF algorithm like PBKDF2, bcrypt, or Argon2, you can reduce the length of the password even further. Using 262,144 (218) hash iterations, you'll increase the security of a 6 word password to log2(77766) + 18 ≈ 96 bits. An example password generated using this method is:
octopus handrail chasing hull shy ambition
That's not hard to remember! It does take some practice and it's not as easy as memorizing a weak password with just one or two words or the name of a pet, but it is something that a child, even a young child, is able to do. And unlike remembering a traditional password where you're out of luck if you forget what special symbol you used or where a character went, a diceware password can be trivially recovered even if some words are spelled wrong (just look in a dictionary).
You can use either the original diceware list or one of the three lists created by the EFF. The benefit of using an EFF wordlist instead of the original is that you can avoid picking obscure words like "ibex" or potentially inappropriate words like "anus", but at the expense of picking words that are longer on average. Simply rolling again when you want a different word is not acceptable because it reduces the keyspace and effectively weakens the password.
Some password managers support multiple equivalent master passwords, making it possible for you to keep a backup password until you are sure your daughter won't forget hers. Then you can revoke your own password so you don't need to have unnecessary access to her passwords.
and she's likely to mislay any physical storage.
If you don't want to synchronize the password database, you can use a stateless password manager. This is a password manager which uses a combination of an identifier for the service you want to log into, as well as a single, strong master password. A stateless password manager works by hashing a concatenation of your master password and the service identifier. It has a few downsides, though:
You can't change a site's password without changing the identifier or master password.
If your master password is ever compromised, so are all your site passwords.
The master password must be strong enough to resist attacks on its own.
If reliance on a storage device to hold the password database is simply unacceptable, then stateless password managers are absolutely the way to go. They can be very secure if used correctly.