What were arguments in favor of the Ptolemaic (earth-centered) cosmos through the Renaissance?
The Ptolemaic system is distinguished not just by being geocentric, but also by its intricate system of epicycles.
The first point about the geocentric system is that it is practically useful. The mathematical calculations used by sailors for celestial navigation, being concerned with the sky as it appears from earth, are still geocentric.
When the Copernican system was invented in the Renaissance, it assumed that planets moved around the sun in circular orbits. This was actually wrong; it was less accurate in predicting planetary movement that the Ptolemaic system, which had planets moving around the earth in complex orbits consisting of a main circular orbit and tiny circles known as epicycles. It was not until Kepler revised the heliocentric system to account for planets moving in elliptical orbits that it actually became as accurate in predicting planetary movements as the Ptolemaic one.