The predicate in your sentence is "is" – not "fascinates".
Let's restructure your thought and write it a different way:
Cats fascinate us for many reasons. One is the popular belief that they have nine lives.
That second sentence is essentially your sentence; however, in your sentence, you've added a clause between the subject (One) and the predicate (is):
One [of the things that fascinates us most about cats] is the popular belief that they have nine lives.
So really your question is about what we should use inside that clause: fascinate or fascinates?
I took a look at some similar sentences, and found examples that suggest it could go either way:
- One of the things that makes mercury dangerous is it can be absorbed through the skin. (Source)
- One of the things that makes that movie [Back to the Future] work is the relationship between Doc and Marty McFly. (Source)
- One of the things that make Barcelona so special is that neighbourhood feeling. (Source)
- One of the things that impact fuel economy is tire pressure. (Source)
One of things that can vex us in English is a phrase like one of the things. Is it singular, or plural? Do we focus on the one, or on the things? This very question was asked on ELU, and a very good answer there shows that there are differing opinions about this. There are many good quotes in that well-referenced answer; I will copy just one of them here:
For most writers the choice depends on whether you're thinking of a single case or a general principle. Usage commentators in the UK and the US have been inclined to say it should be the plural; and the Harper-Heritage usage panel voted heavily in its favor (78%). Yet Webster's English Usage (1989) found ample American evidence for the singular construction, and it's just as common as the plural in British data from the BNC. Writers using the singular take their cue from one, whereas plural users are responding to those [people] or the [things]. (Pam Peters, The Cambridge Guide to English Usage)
As for the sentence you proposed:
One that fascinates us most about cats of the things is the popular belief that they have nine lives.
That doesn't work. However, you could rearrange it so that it would work just fine:
One thing that fascinates us most about cats is the popular belief that they have nine lives
Changing one of the things to one thing is actually a good way to dodge this problem. It reads nicely, and doesn't really shift the meaning that much:
- One thing that makes mercury dangerous is it can be be absorbed through the skin.