This probably is not the sort of thing you find being debated at the U.N., but with summer coming up it's a good question! If one way of doing it is better than the other, it could save time for millions of people!
One thing that complicates the answer is the variables. You have the speed of the mower, the width of the mower, the dimensions of the yard, the amount of time it takes to turn, etc. There's also the issue of things like trees and shrubs in the yard, which might make one or the other pattern better. Let's ignore trees and shrubs, assume that the mower is 2 feet wide and assume a square yard that is 100 feet on a side (making it 0.25 acres or so). Here are the details of your paths:
If you go ahead and calculate the distance each of you have to travel, it is the same -- both of you have to walk 5,000 feet or so (nearly a mile!). The big difference comes in the turns. Your brother has to make 49 180-degree turns. You, on the other hand, have to make 99 90-degree turns. If you assume that it takes twice as long to turn 180 degrees than it does to turn 90 degrees, then the two paths are approximately equal again.
- Your brother would walk 100 feet up one edge of the yard. He would then make a 180 degree turn. Then he would walk back another 100 feet, make a 180 degree turn..., and so on, 50 times.
- You, on the other hand, would walk 100 feet up one edge, turn 90 degrees, walk 98 feet, turn 90 degrees..., and so on, spiraling in toward the center. You would have to make 25 spirals, with each spiral getting progressively smaller.
If one type of turn is significantly more involved than another, then that could shift the equation. For example, on a riding mower, a 90 degree turn is probably much easier than a 180. With a walk-behind mower, a 180 and a 90-degree turn may differ only slightly.
So the key is to get out there with a stop watch and see which type of turn is more efficient for you. Or get a goat. Or get one of the new robot lawn mowers!