Ecuador is a country named after an imaginary line. In Spanish, ecuador is simply the ordinary word for the equator, and the equator passes very close to the capital, Quito.
The line's own name is older and stranger. Etymonline traces equator to the Medieval Latin phrase aequator diei et noctis, "equalizer of day and night." It is built on the Latin verb aequare, "to make equal." The first sense was astronomical: when the sun sits on the celestial equator, day and night are the same length everywhere. English applied the word to the line around the Earth by the 1610s.
For most of its colonial history the region had no such name. From 1563 it was run from Quito as a Spanish real audiencia, an administrative district, and Etymonline notes that before 1830 it was known by that chief city.
Science helped change that. In 1735 a French expedition set out to measure an arc of the Earth's curve near the equator, reaching Quito in June 1736. Led by Charles Marie de La Condamine, Pierre Bouguer and Louis Godin, it worked alongside a Lapland expedition. Together they showed that the planet is flattened at the poles. The mission made the "lands of the equator" famous in Europe.
After independence from Spain, the area joined Gran Colombia. In 1824 Gran Colombia redrew its internal map, and the district around Quito became the Ecuador Department. When the region broke away from Gran Colombia in 1830, it kept that name and became the República del Ecuador. Wikipedia's account of the French expedition says its fame influenced the choice, though it frames this as an influence rather than the whole reason.
Under Spain, the district had been identified by a city. As a republic, it was named for a piece of global geometry.
So the name stacks two ideas. The country is named for a line you cannot see, and that line is named for the days when light and darkness are shared equally.