What visible star is closest to the ecliptic? If we were to project the ecliptic out as an infinite plane, what visible star (besides the sun) would lie closest to it?
Regulus ($\alpha$ Leo) is the one to beat, with ecliptic latitude $\beta$ = +0.46$^\circ$ and apparent V magnitude +1.4.
Some dimmer stars are closer to the ecliptic, e.g.
$\alpha$ Lib ($\beta$=+0.33$^\circ$, V=+2.7),
$\delta$ Gem ($\beta$=-0.18$^\circ$, V=+3.5), and
$\delta$ Cnc ($\beta$=+0.08$^\circ$, V=+3.9).
For more suggestions, poke around in Stellarium with the Ecliptic turned on.

If you want linear distance from the ecliptic plane, multiply the distance from us by $\sin \beta$.