Where was Ravana born? Do we have any mention of Ravana's birthplace in any Ramayana or Purana?
Ravan was born most likely in Ashram of Vishrava or else Rasatal.
Ravan's maternal grandfather Sumali hid in Rasatal after bitter fight with Vishnu in which they lost badly. Lanka, which was their home then, was given to Kuber. When Ravan was born, Rakshash were still living in Rasatal.
However, Kaikasi, mother of Ravan, may still be living in Ashram of Vishrava muni at the time of birth of Ravan. Kaikasi had visited Vishrava muni in desire of sons as great as Kuber, following which Vishrava muni gave her boons of four children. Although it is not stated clearly if she left for Rasatal after receiving boons.
It's likely that she continued to stay with Vishrava Muni. This may be confirmed with following incident. When Kuber (living in Lanka) came to visit his father Vishrava, Kaikasi spots him in the ashram and exhorts Ravan to be as great as him.
There is not much information about ashram of rishi Vishrava. However, he was highly influenced by his father, rishi Pulastya. Likely that he lived nearby. Rishi Pulastya lived at the ashram of rishi Trinbindu, near great mountain Meru. He was married to daughter of rishi Trinbindu, and likely their son rishi Vishrava was born there as well.
As the other answer states Ravan was most likely born in the Ashram of Rishi Vishravas since his older step-brother Kuber was also born there. I am sharing the information from the Mahabharat about the location of the Ashram.
"Dhaumya continued, 'I shall describe to thee those sacred spots capable of producing merit that lie on the west, in the country of the Anarttas, O Bharata, there, flows in a westward course the sacred river Narmada, graced by Priyangu and mango trees, and engarlanded with thickest of canes.
All the tirthas and sacred spots, and rivers and woods and foremost of mountains that are in the three worlds, all the gods with the Grandsire, along with the Siddhas, the Rishis and the Charanas, O best of the Kurus, always come, O Bharata, to bathe in the sacred waters of the Narmada.
And it hath been heard by us that the sacred asylum of the Muni Visravas, had stood there, and that there was born the lord of treasures, Kuvera, having men for his vehicles.
There also is that foremost of hills, the sacred and auspicious Vaidurya peak abounding with trees that are green and which are always graced with fruit and flowers. O lord of the earth, on the top of that mountain is a sacred tank decked with full-blown lotus and resorted to by the gods and the Gandharvas. Many are the wonders, O mighty monarch, that may be seen on that sacred mountain which is like unto heaven itself and which is visited by celestial Rishis. There, O subjugator of hostile cities, is the sacred river called Viswamitra belonging to the royal sage of that name and which abounds, O king, in many sacred tirthas.
The description of the environs and landmarks clearly points towards the erstwhile Anarta Kingdom of Gujarat and the river Vishwamitri still flows in that region.