Just performed some experiments. I used a .5 liter disposable drinking water bottle and cool tap water (I didn’t standardize the water temp, but it was all drawn from the cold water tap on the upper floor of a tall building, so it shouldn’t vary much). I dropped one 2.5g sugar cube in the bottle (I used a sugar cube because I could easily see whether it had dissolved or not without disrupting the agitation), and then agitated either side to side (axially to my forearm) or up and down, and timed how long it took for the cube to completely disappear.
If the bottle was completely full of water (no air space), side to side was the clear winner. It took 100 seconds vs. 120 for up and down.
If there was an air space in the bottle (100cc), the mixing was greatly accelerated and the difference between the two methods decreased, but side to side was still slightly faster (70 seconds vs. 80 seconds).
So the most efficient method would appear to be to leave an air space in the bottle and agitate by rotating your forearm on its longitudinal axis.