How can I make a button trigger a relay that stays on temporarily without using an Arduino?

One solid approach is to use a capacitor to store charge when you press the button. This charge can then power a MOSFET that keeps the relay activated for a predetermined time, based on how you size the capacitor and resistor. Check out this example circuit I found—looks straightforward!

You can easily set up a relay to latch itself using just some basic wiring. Make it so the relay keeps getting power when it's activated, and use another button to interrupt that power when you need it to turn off. No fancy chips required for a simple on/off toggle!

You might want to check out time delay relays. They're commonly used in industrial applications and typically have a knob for setting the duration. Just keep in mind these can be pricier than an Arduino option!