One of the things that I hate is that some people like to spread rumors

"One of the things that I hate is that some people like to spread rumors."

"One of the things that I hate is some people like to spread rumors."

Your sentences can be rearranged as That some people like to spread rumors is one of the things that I hate.

But would be wrong as * Some people like to spread rumors is one of the things that I hate.

If we look at both thats:

“that I hate” is a relative clause modifying “things”. In relative clauses, “that” can changed for “which” and can be omitted, e.g. “The hat [that/which] I bought yesterday has been stolen.”

In “A modern English grammar on historical principles Part3 Syntax V2 1927” Otto Jesperson introduced the idea of a “content clause” – these have some similarities to noun/relative clauses but also some distinctions.

“that some people like to spread rumors” is a content clause. In content clauses, “that” is not usually omitted, and “which” cannot substitute for “that”, e.g.

“That he had stolen the hat (content clause) was denied.”

but

“He said ‘I have not stolen a hat’ (direct quote as object)” is correct.

And

He said which he had not stolen a hat –is clearly wrong.

The necessity of “that” in content clauses is clearer when the content clause acts as a primary (the subject or agent):

“That he had no money meant he could not buy the hat.” –you will note that that cannot be omitted.

Content clauses are often found supported by “the fact”, e.g. “The fact that he had stolen the hat was denied.” This now changes the content clause (that he had stolen the hat) into a defining relative clause - “The fact [that] he had stolen the hat was not denied”

In your example,

"One of the things that (OPTIONAL) I hate is that (PREFERABLE) some people like to spread rumors."

And

"That (NECESSARY) some people like to spread rumors is one of the things that (OPTIONAL) I hate."

And "The fact that (OPTIONAL) some people like to spread rumors is one of the things that (OPTIONAL) I hate."