Declining Sea Levels - Benoit de Maillet, a French anthropologist who lived from 1656-1738 and guessed (incorrectly) that fossils at high elevations meant Earth was once covered by a large ocean. This ocean had taken 2 billion years to evaporate to current sea levels. Scientists abandoned this when they realized that sea levels naturally rise and fall.
Cooling of the Earth - William Thompson, later known as Lord Kelvin, assumed that the Earth was once a molten ball of rock with the same temperature of the Sun, and then has been cooling ever since. Based on these assumptions, Thompson calculated that the Earth took somewhere between 20 and 400 million years to cool to its current temperature. Of course, Thompson made several inaccurate assumptions, about the temperature of the Sun (it's really 15 million degrees Kelvin at its core), the temperature of the Earth (with its molten core) and how the Sun is made of hydrogen and the Earth is made of rock and metal.
Cooling of the Sun - In 1856, the German physicist Hermann Ludwig Ferdinand von Helmholtz attempted to calculate the age of the Earth by the cooling of the Sun. He calculated that the Sun would have taken 22 million years to condense down to its current diameter and temperature from a diffuse cloud of gas and dust. Although this was inaccurate, Helmholtz correctly identified that the source of the Sun's heat was driven by gravitational contraction.
Rock Erosion - In his book, The Origin of Species by Means of Natural Selection, Charles Darwin proposed that the erosion of chalk deposits might allow for a calculation of the minimum age of the planet. Darwin estimated that a chalk formation in the Weald region of England might have taken 300 million years to weather to its current form.
Orbit of the Moon - George Darwin, the son of Charles Darwin, guessed that the Moon might have been formed out of the Earth, and drifted out to its current location. The fission theory proposed that the Earth's rapid rotation caused a chunk of the planet to spin off into space. Darwin calculated that it had taken the Moon at least 56 million years to reach its current distance from Earth. We now know the Moon was probably formed when a Mars-sized object smashed into the Earth billions of years ago.
Salinity of the Ocean - In 1715, the famous astronomer Edmund Halley proposed that the salinity of the oceans could be used to estimate the age of the planet. Halley observed that oceans and lakes fed by streams were constantly receiving more salt, which then stuck around as the water evaporated. Over time, the water would be come saltier and saltier, allowing an estimate of how long this process has been going on. Various geologists used this method to guess that the Earth was between 80 and 150 million years old. This method was flawed because scientists didn't realize that geologic processes are extracting salt out of the water as well.