Extenders are a pain to mess with since you end up with two network names. Also they are only as good as the signal received. If it is a bad signal received it will only transmit a bad signal. You can mess with placement to find the optimal location. Still it's a pain and there are better solutions.
Powerline Networking:
This is usually the best easy and budget friendly option. Your home electrical wires are used for networking. You just need adapters at the source (router) for creating the network. Then you need adapters at the destination. These adapters can be WiFi access point, Ethernet adapters or a combo of the two.
Using it for WiFi. As long as you configure all the name and security settings exactly the same between the Powerline Access point and your router. Then you can just connect to the best signal without selecting networks each time.
The biggest caveat of Powerline is not all homes handle it well. You can only try it and find out. Sometimes there is some device which causes interference in your powerlines. If you have a huge house or some other distance issue (detached garage). The powerline runs may be too long.
Mesh:
Mesh has an issue like that of range extenders. Being that each node can only be as good as the signal it receives. The advantages being you can easily add nodes to reduce signal losses. Also they all use the same network info. Making moving between nodes easy like Powerline or any network using access points.
It has disadvantages. Mesh networks generally cost more money compared to other options especially if you need to start adding nodes or go with high performance. The Mesh adapters also need to act as the router. Which could mean a lot or reworking or loss of advanced networking capabilities. Depending on how you are setup right now. For most people it wouldn't really matter.
Most use phone apps exclusively for configuration. That means the manufacturer probably gets all your info. Also if you like the good ole' web interface for configuring your router. That isn't an option in a lot of instances.
Ethernet + Access Point: This is what I personally use.
Similar to Powerline networking with access points. The big difference is you are using Cat 5/6/7 cables for your wired infrastructure rather than powerlines. That means longer runs. No real speed loss by length. Little likelihood of problems from interference. You also can basically use any wired access point you want or configure any router you want to act as access points. This is the most powerful option given all the high performance equipment you can install with a 1 GbE or even 10 GbE backbone. I mean you can have even faster fiber optic connections. But that gets ridiculous for home users.
The biggest downside being complexity. If you don't have Ethernet cables already. You have to run them. Configuring routers to act as access points is generally more complex a procedure too.
MoCA:
Pretty much the same as powerline adapters. Except you use existing RG-6 cables (cable TV). You don't have as many options but there are less interference issues. A big caveat being. They can cause serious issues with some TV providers. Depending on how their devices communicate with each other. Mainly being some use their own competing communication between devices in your house over MoCA.
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TLDR
Unless someone already has Ethernet run. My first choice is to install a Powerline WiFi adapter for them. Then see how it works out. If the house ends up being a problem. You just send it back.
This is my preferred kit but availability can be tough. (TP-Link Tl-WPA8630 V2 KIT)
AC1350 Wi-Fi range extender and Av1300 power line Edition the terms “ av1200” and “ 1200mbps” are derived from applicable specifications and refer to the theoretical maximum physical-layer data transfer rate. Those terms are used on this product solely to indicate its compatibility with other “ a...
www.amazon.com
This is my alternative if the first can't be found. (TP-Link Tl-WPA7510 KIT)
TP-Link AV1000 Powerline WiFi Extender(TL-WPA7510 KIT)- Powerline Adapter with Dual Band WiFi, Gigabit Port, Plug&Play, Power Saving, Ideal for Smart TV, Online Gaming
www.amazon.com