It's a well known fact that the Greek Gods resided in the fabled mountain of Olympus. However most mythology was created based on factual stories that have been twisted or altered. Why is Mount Olympus so important to the Greeks that the Gods would live there?
The Olympians defeated the Titans on Mt. Olympus and decided to build their kingdom at the scene of their greatest victory.
The Titanomachy (the great war between the Olympians and the Titans) took place on Olympus. After the Olympians defeated the Titans, they proceeded to build their kingdom there, in honour of their victory.
The exact mountain range referred to by Homer is still unknown as he offers little to no geological information about it.
But there are some theories that the Greeks assumed Olympus was the seat of power for their gods as it was so high it touched the clouds.
If this is true, then of all of the mountain ranges called Olympus throughout ancient Greece, Turkey and Cyprus only one breaks the 2000 meter trehsold to reach the clouds : Mytikas at 2918 meters (9570 feet).
Olympus has always been the home of the gods, even before Zeus' reign:
(ll. 109-120) First of all the deathless gods who dwell on Olympus made a golden race of mortal men who lived in the time of Cronos when he was reigning in heaven. And they lived like gods without sorrow of heart, remote and free from toil and grief: miserable age rested not on them; but with legs and arms never failing they made merry with feasting beyond the reach of all evils. When they died, it was as though they were overcome with sleep, and they had all good things; for the fruitful earth unforced bare them fruit abundantly and without stint. They dwelt in ease and peace upon their lands with many good things, rich in flocks and loved by the blessed gods.
(ll. 121-139) But after earth had covered this generation -- they are called pure spirits dwelling on the earth, and are kindly, delivering from harm, and guardians of mortal men; for they roam everywhere over the earth, clothed in mist and keep watch on judgements and cruel deeds, givers of wealth; for this royal right also they received; -- then they who dwell on Olympus made a second generation which was of silver and less noble by far. It was like the golden race neither in body nor in spirit. A child was brought up at his good mother's side an hundred years, an utter simpleton, playing childishly in his own home. But when they were full grown and were come to the full measure of their prime, they lived only a little time in sorrow because of their foolishness, for they could not keep from sinning and from wronging one another, nor would they serve the immortals, nor sacrifice on the holy altars of the blessed ones as it is right for men to do wherever they dwell. Then Zeus the son of Cronos was angry and put them away, because they would not give honour to the blessed gods who live on Olympus.
I'm not aware of any "in-universe" explanation of why the gods would favour Olympus over any other place. The logical explanation would be that it's the highest mountain in Greece, the closest you can get to the heavens.