I think left side makes more sense (most people get on-off bikes and horses from that side), but right is not too bad either, as long as one standard is set globally.
The British rode horses on the left, because it allowed the rider to wield a sword with their right hand. We exported this to the British Empire and many of those countries still follow our practice.
The Americans used the right hand side because, when they used wagons for transport, the driver would sit on the left such that, if two wagons were passing each other, the drivers could easily look down to their left and check the wheels weren't going to clash. Also they would sit on the left rear horse, because that meant that they could use the driving whip on the horses on the left or the right of the team without having to reach across.
Supposedly Europe drives on the right because Napoleon was left handed, so he made everyone else follow him after he conquered much of Europe, but there's no actual documentary proof of this.
The last European country to change from left to right was Iceland in 1968. These days it would be prohibitively expensive to change over a modern, industrialised country from left to right hand traffic.