Wireless extenders aren't going to be much more expensive than running a cable out there, so it's just a matter of preference. The wired connection will give you better security, since no one will be able to jack the signal from your garden, but the wireless option won't require you to run a cable through it. Depending on how close to the office the router currently is, you might need to run a cable all the way. If it's at the very limit (meaning that you couldn't move it any closer without moving it into the actual garden), an antenna might be more than you want to spend (because you would need one rated for outdoor operation, which are more expensive).
A 100ft ethernet cable and a 4-port network switch won't run you that much. You can probably get a cable in green too, so if you have somewhere out of the way to put it, it'll blend in even better. Some of the cheaper indoor extenders go for about the same amount as the cable + switch, but outdoor ones start at about double that.
I suppose it's also possible to get an antenna with a cable long enough to reach from the router to inside the office, if you look around, but at this point you should just get an access point.
Edit: No. You would keep the router that you have in the house and run the ethernet cable to a wireless access point that has no additional routing functionality. You do not want to run a router to another router unless you specifically need to do so, because it will make your network setup more complicated that way. If you must use a router, some can be put into access / repeater mode and it should disable features like DHCP which your primary router will be handling. (You may need special firmware like DD-WRT for this.)