To get this information, the captain needed a clock which kept accurate time for a many months: it could be set in Greenwich (or set to Greenwich time at a location of known longitude), and used later to give "Greenwich time" of local noon. Such clocks ("chronometers") were in fact developed in the 1700s, but clocks of the 1600s were not accurate enough, especially on a ship that rolled and pitched, and their errors accumulated rapidly.
A less precise clock may be used, if somehow it can be constantly corrected, reset to the correct "Greenwich time" at frequent intervals. In a later era this was done using time signals obtained by radio, but in the 1600s accurately timed celestial phenomena held the greatest promise. One class of such phenomena were the eclipses of the four large moons of Jupiter, discovered by Galileo and easily seen through even a small telescope.
In particular, Io, the innermost moon of Jupiter, seemed suitable: being closest to Jupiter, Kepler's 3rd law assured that it had the fastest motion, making its entry into eclipses and out of them particularly rapid. With an orbital period of 1.77 days, Io also offered the largest number of eclipses, and every one of its orbits crossed Jupiter's shadow. (In the satellite age Io was found to have other unique features, such as sulfur volcanoes.)
Giovanni Domenico Cassini, an Italian astronomer who headed of the Paris Observatory, therefore assigned Rømer to make a table of the predicted times of Io eclipses, allowing sailors at sea to set their clocks (within a minute or so, deemed accurate enough). Rømer did so, but soon discovered that the period was not constant. When Earth (which moves faster than Jupiter) was approaching Jupiter, the observed period was shorter, and when it was receding, longer.
He guessed the reason: light did not spread instantly, but (like sound) did so at a certain speed. If Earth and Jupiter maintained a constant distance, the eclipses would have been spaced at regular intervals, equal to the orbital period of Io. When Earth is approaching, however, the return trip is shortened, compared to the time it would have taken if the distance stayed constant. When Earth is receding, the return trip is longer, and the time between eclipses is longer too
That gave Rømer convincing evidence that light spread in space with a certain velocity--later denoted by the letter c (lower case, not capital). However, he and his contemporaries had only a vague idea how big c was, because the dimensions of the solar system were uncertain. About that same time, the French astronomer Jean Richer used a telescope to estimate of the distance of Mars, and gradually, the value of c was obtained with increasing accuracy. Today it is known to an accuracy of 9 decimals, and has therefore been used to define the metre, the unit of length, replacing optical wavelengths or scratches on a metal bar kept in a vault (supposedly derived from the size of our globe).
And the problem of longitude?
It turned out that observing the eclipses of Io from a constantly moving ship, even in a calm sea, was a difficult task. Even a small telescope magnifies all motions tremendously, and early telescopes in particular showed only a small patch of the sky. Also, the method required a sky free of clouds. On the other hand, the method proved very useful for determining the longitude of ports, capes, islands and other features on land.
Consistent determinations of longitude from a moving ship had to wait for sophisticated clocks, using a balance wheel compensated for changes due to variation of temperature. One early model of such a "chronometer" accompanied Captain James Cook on his journey around the world.
See "Ole Rømer, the speed of light, the apparent period of Io, The Doppler effect and the dynamics of Earth and Jupiter" by James H. Shea, American Journal of Physics, 66, July 1998, p. 561-9