One of the things that makes him great is he brings it every night.
I'm pretty sure it should be that make him in the plural, because one of the things is referring to a lot of things and a lot of things should take a plural verb. But I always hear native speakers in America say it in the singular, so with an s.
Another example:
This is one of the responsibilities that comes with greatness and he understands that.
Native speakers always use the verb in the singular (here, comes) even though [I believe] it is grammatically incorrect to use the singular form of the verb there.
Best Answer
Some of the style guides that I have are categorical in claiming that the plural verb is correct in such constructions as:
Follet in Modern American Usage (p298) states:
Partridge in Usage and Abusage (p214) states:
The Right Word at the Right Time (p405) states:
Garner in Modern American Usage (p590) writes:
Other style guides offer a more nuanced approach. For example, the conclusion after a lengthy discussion of the construction in Merriam Webster's Dictionary of English Usage (p690) is:
Peters in The Cambridge Guide to English Usage (p394) states:
Finally, Steven Pinker, in his recent The Sense of Style (p250) writes:
It is interesting that one of the sentences that The Right Word at the Right Time lists as ungrammatical is the following:
This sentence was written by pre-eminent British linguist Professor David Crystal.