I had recently purchased a refurbished T61. I also bought a Wifi adapter, though I have Wi-Fi at home and do not expect to travel much with the laptop, because I thought that WIndows-based laptops from that time period needed one. I was, however, able to connect with my own WiFi without the adapter; should I return the adaptor or are there security or other benefits to using one?
The T61 comes with built-in Wi-Fi support. Almost all Wi-Fi routers can recognize and work with older WiFi adapters although the speeds and security levels may not be up to today's norms.
Nevertheless, for home use that's probably not a concern.
For maximum speed and security consider running an Ethernet cable from the router to the T61. Of course if you need portability within your house then this is not an option.
P.S. Almost all notebooks of the T61's generation came with built-in WiFi. If you replace the T61 with something similar then it's 99.9% likely to have Wi-Fi. You can safely return the separate WiFi adapter.