Really?
You want him to buy a 40$ cable (and he has 2 garages) without any access point as compared to a 100$ to 150$ fast router? perfect solution!
I agree we need more info and it might be that his router is old and spending $100+ on a new one would help. However, its not going to work miracles.
I'm just going by experience. I've been a computer network systems engineer for 20+ years. 100 feet in one direction through exterior walls and then into a steel shed and 75 feet in another direction to the garage through multiple walls and furniture..... It might function, but it won't be great unless you're using commercial grade equipment.
If the router is old, a new router might be good enough by itself, but it will have to be a high power one with multiple antennas, internal or external. Simplest solution, but $$$ unless he was wanting a new router anyway.
First step would definitely be to try moving the current router as close to a central point as possible and near an outside wall in the back of the house if that's where the shed is.
Install that free WiFi Analyzer metering app on your phone to help you find the ideal spot to maximize signal in all locations. It's not complicated to use.
If you still can't get enough signal, go to BB or WalMart and get a Netgear N750 router for $100, or more powerful make/model if you can afford it. Make sure you find out how long you have to return it for a refund. Install the new router and test again with the WiFi metering app. You'll probably see improvement.
If still not enough, take it back and put your money towards tried and true hard wire and extenders.
Bulk Cat5 wire at Lowe's or Home Depot is dirt cheap but you'd need the ends and termination tool. If your cable installer is running your line, give him a couple of beers and he'll probably take care of the cat5 ends. Or use a pre-made line. 100' cat5e Mediabridge brand (good) cable is only $13. That's probably cheaper than bulk and the ends are installed. You don't need cat6 for what you're doing.
You might find that you can move your current router close enough to the garage or shed to get acceptable signal to that location. Then you might get by only having to run wire to the other location and use one extender.