Generally, main factors in the population size of a place were how much food it could produce, how remote it was and how economically important it was. Worldwide, like other answers have pointed out, population increased very gradually until a dramatic increase in 17th-18th centuries thanks to many advances in adricultural tools and techniques, and a really really dramatic increase in mid 20th century due to mechanisation of agriculture and increased use of chamical fertilizer.
Of course the size and terrain of the island will be important, as will be fishing resources around it.
The main economic factor for your island nation would probably be naval trade. This would depend on natural resources or proximity to other big and prosperous population centers. An island that is economically important would have bigger cities and so more population.
Economic importance of the island could hugely increase and decline with changing technology. For example an island that used to be hugely important as a stopover port along an important trade route could become much less important after ship range increased or a canal has been dug. Or if the island provided a very scarce resource, but later more plentiful supplies of that resource started coming from newly discovered (for that part of the world) far-away lands. The opposite could also happen, when a resource that used to be unknown, unimportant or previously impossible to mine has become available (e.g. discovery of oil on the island) and the island suddenly became more economically important. Or perhaps its beaches and wildlife suddenly turned it into a major tourist attraction - that would have a dramatic effect on the population size.
Of course, natural disasters, war and epidemics would have a huge impact on the islang population. After a cataclysmic event the population could eventually recover, or it could never return to previous levels.
It's hard to say any more without knowing the history of your world and especially the part of the world where the island is.