The first NES version's mine elevators are activated not by an item but by four separate machines that receive a random on/off setting when you enter from above and can be switched by pressing the B button on them, but switching one also switches the two machines beside it, with a wraparound effect where the machines on the left and right ends are considered to be "beside" each other. Since another B button press on the same machine reverses its entire effect, you should never have to press B on a single machine more than once to get them all on. You can have fun solving this casually, but in a speedrun you won't want to have to stop and think too much, so it can be better to memorize what to do for each scenario you find the machines in. It is possible to stumble upon all 4 machines already lit up when you enter! But since that's unlikely, you should always scroll the fourth machine onscreen and make decisions like so:

No lights already on: Press all 4 buttons. 1 light already on: Press the button that's 2 machines away from the one that's already on. (This puts you in the "center" of the three unlit ones.) 2 lights already on and are beside each other (or on opposite ends): Press both buttons that were already on to begin with. 2 lights already on and are NOT beside each other in any way: Press both buttons on the machines that were OFF to begin with. 3 lights already on: Press the one that was off and the two on either side of it--that is, press all EXCEPT the machine in the "center" of the lit ones.

If ever you get confused and fail to recognize what to do next, don't panic; just toggle lights until you get a pattern you do recognize and work from there. If you're really going for speed, though, be sure to go to the rightmost switch that needs toggling before you work your way back to the left. Be mindful of where you do the toggling: On the right side of a machine, you need the left half of Dizzy's whole body in front of it, but on the left side of a machine, just having his whole right hand in front is enough.

And remember, you only need all four elevators on when you're ready to visit the dragon. If you only want to ride the minecart, just get machine #1 on and ignore the rest. Be careful at the mine entrance, too, because merely jumping from on top of the box is enough to flick the screen and change all the machine settings.