Did scientists know about harmonic oscillation? Looking back from the present, are there any theories about how timekeeping could have been made more efficient (not even necessarily at sea) with technology that existed in the fifteenth century?
No. Basically, a clock is a series of gears: driving force at one end; alternating catch , the escapement, at the other. For a driving force, the 15th c. clockmakers could use a spring, or a weight. Both of these had problems- a hanging weight on a sailing ship could be flung around, changing the force. A spring wouldn't be flung around, but it would have the most force at the beginning, when fully wound, and gradually become weaker as it unwound- so the gears would also have varying amounts of force over time.
The only alternating catch used in the 15th c. was the verge and foliot escapement. Its resonance would change with not only the variation of force applied to the gears, from the spring or the weight, but also the small weights on the end of the arms of the foliot would be affected by the movement of a ship, as well as how much friction was in the escapement arms, bearings.
Some of these variations would tend to cancel out- the clock would have an average time, despite irregularities in beat. But about the best that one would do on land was keep within about 15 minutes, for every 24 hours. For a ship cruising very,very smoothly at 10 knots, that translates to 2.5 nautical miles per day of error. In a voyage of only 10 days, therefore , even the best clock would have an error of 25 miles: that could put land out of sight of a lookout, or put the ship on a reef.
A few inventions made things better. A cross-beat escapement was invented by Buergi in 1584: with two escapement arms, a lot of error was reduced. He also invented the remontoir, a way to distribute the force of the mainspring over time by having it drive another, smaller spring. Then later in the 17th c., Galileo noticed that pendulums beat extremely evenly. Although he himself didn't add them to clocks, others did ( like Huygens) . But still, even an error of a few minutes a day was critical. Not just because a navigator on board a ship needed a good chronometer to carefully place his position on a map, but because the cartographer needed a good chronometer in order to correctly draw the map the navigator was using. Once a cartographer was sitting on a land, he could take the precise astronomical sightings to measure the longitude...on a pitching ship, that wasn't possible.
It would take a lot of patient perfection of gear teeth, pivots, control of thermal expansion, etc. through the 17th c., before a practical chronometer was created by Harrison, in the earlier 18th c.
Landes: Revolution in Time