Basic design considerations:

• Needs to be big enough to contain some cool goodies for kids (currently 7, 4, and 1 years old). So big enough for some sweets, biscuits, lego, etc. And to hold the electronics, display, and keypad.
• Needs to be robust enough to withstand said children playing with it, but doesn't need to be high security.
• Needs a screen and keypad to present questions and accept answers. Numeric keypad is a minimum; realistically, a full keyboard isn't likely to fit.
• Needs to be easy to program to update the questions; OTA is a pretty much a must; I'll never bother to use it if I have to re-flash it with new questions. I guess telnet over FTDI could work at a minimum, but an HTTP interface would be ideal.
• Ideally, should pass the 'wife test' – can the (intelligent but non-engineer) wife set questions on it? (i.e. can't be too complex)
• Should be easily hackable to make the questions more complex. If it's got wifi for OTA, it could take part in a larger connected challenge.

So it looks like a pre-made box is preferable; my woodwork skills suck, and it'll provide a solid, robust starting point.

ESP8266 provides an OTA-updatable controller.

Screen requirements aren't huge; pose a simple question, and ideally some space for a progress display. I've got a Nokia 5110 screen module lying around, so that'll do for a screen.

Keypad – I had an MPR121 module to hand. Initially I imagined using this to create a keypad with nails or similar stuck through the wooden box & sanded flat, but from a little reading it became apparent quickly that designing a capacitive keypad is a small artform in itself, so for now I've gone for a pre-made keypad.

Espruino may not be the optimal way to program microcontrollers, but it does provide an easy way to update portions of the code (the questions), and js is very user-friendly. It also provides robust maths and string-handling features.