Nearly all scientific development happens through observation followed by correlation and then a search for mechanism.

The lunar connection to the tides goes way back. Seneca in De Providentia (On Providence) writes in the first century:

anyone who has watched how the shore is laid bare by the retreat of the sea into itself, and how within a short time it is again covered, will believe that it is in obedience to some hidden law of change that the waves are at one time contracted and driven inwards, at another burst forth and regain their bed with a strong current, since all the while they wax in regular proportion, and come up at their appointed day and hour greater or less, according as the moon, at whose pleasure the ocean flows, draws them.

It must have been widely known, since he is using it to support his stoic concepts rather than introducing it as its own piece of knowledge.

Bede also wrote about the tides in 725 AD (On the Reckoning of Time, Ch 29) goes on to talk about neap tides and spring tides. He had the advantage over Seneca that the atlantic tide is 5-10X the size of the mediterranean. He even went on to speculate on a mechanism, estimating that the rivers produced less water during waxing crescent and waning gibbous moons. This is the wrong mechanism, of course, but it indicates that the monthly pattern was clearly recognized.