To my understanding, Earth’s Moon’s surface turns red in a lunar eclipse due to refraction of light through Earth’s atmosphere. When Saturn has a lunar eclipse with Titan (so Saturn rests between Titan and the Sun, to clarify) would Titan’s surface change color? If so, what color would it become? Would this effect last much longer than it would on Earth?
It would be dark.
Titan in eclipse can be dimly lit by refracted sunlight and light scattered by Saturn's rings. The refracted light would be reddened, but the scattered light would be white.
But there isn't much light that far out, and so the brightness would be very low. It has been imaged in eclipse: See
The ring-light would be brighter and so Titan would be illuminated principally by the white light scattered by the rings, and so it wouldn't appear to change colour very much, only become much dimmer. A long exposure image was required to capture the image on the linked page.
From above Titan's clouds you would see the planet as a huge red ring, with two white streaks on either side (the rings, viewed nearly edge on)
Because of the much greater distance from the sun, and the much greater size of Saturn compared to Earth, while there may be a brief period when Titan goes into eclipse where there will still be some refracted sunlight, as well as some reflected light from the rings, for the majority of the eclipse Titan will be effectively dark.
And this assumes you have an observer above Saturn's atmosphere, in order to have the equivalent view...