Amplifying the WiFi signal
An Access Point would be sufficient - &amp; cheapest. 

[just some examples from Amazon, no recommendations specifically. Note, the filter on Amazon isn't great, there are other device types in there, as well as simple Access Points]

An Access Point needs no additional functionality other than being able to push a signal further.
You wire it via your existing Ethernet to where the signal will reach the garden. All routing/DHCP functionality is then still provided by your existing router.  

If you can't, for any reason, get an Ethernet cable far enough to reach, then you would need a Range Extender, which is in essence the same thing, but wireless throughout. I'd still go for an AP as first choice, if at all possible.