You want to buy a wifi router for mining. You would want to buy a super stable wifi router that do not need to power cycle every now and again. I would not buy a second hand router since a brand new good router is not expensive anyway and will last for a few years.
Broadband Router with 4-Port Switch
Model: WRT54GL
This is the router I have been using for 4 years. Super reliable. Only get it if it is brand new. Don't buy it secondhand.
I have one WRT that has lasted a very long time pretty darn impressive. I personally like having additional options on my routers I currently am running ddwrt and tomato on different routers.
With OP not having a ethernet connection and only wifi. I would suggest a router with client mode that you can connect to wifi from it and plug ethernet cords into it for internet connection.