What kind of wire is best to use for connecting a FM radio to the antenna? This article doesn't say what kind of wire is used.
To construct an antenna, the focus should be on the resonant length, not on the wire used. It needs to be conductive; if it's going to be outside, protected against the elements, but if you're using free wire, then you can replace it after it rusts through.
Antennas can be either connected directly to a radio, or via some feed mechanism. Depending on which frequencies you're wanting to use (you say FM, but that's not a frequency, it's a mode), you may well find that a simple coat-hanger is more than sufficient.
That article is terribly written, because it doesn't tell you how to construct the dipole part of the antenna and the pictures make it look like you're modifying something that already exists, but it does tell you what to use: 300 Ω twin-lead. This is a standard cable for connecting TV antennas (though less common than coaxial cable these days) and should be easy to find at stores that sell TV antenna components.
Also, this is not answering the question you asked, but if you are building an antenna design that uses a single strand of wire, not twin-lead, then the type of wire basically doesn't matter except for non-electrical properties (whether it is strong enough, whether it will corrode outdoors).