The second one is the better choice, because does agrees with one. Here, men is the target of the preposition of. The phrase of the men serves to qualify one; it is not the subject of the clause.
If you remove of the men, you can see that does is correct (though one then needs a specifier; I add the to fulfil this requirement):
He is the one who does the work.
The emphasis here is on a single man. Though he is not the only one who does the work, he is the subject of discussion. The sentence isn't talking about a group of men; it's talking about one man who also happens to be part of a group. Thus, do should be conjugated to agree with one.
Both sentences can be correct:
This is one of the things which is really important to her.
This is one thing. It is really important to her.
This is one of the things which are really important to her.
Out of all of the things that are really important to her, this is one.
So, either one thing is important, or the things are important. In the singular meaning, the other things might or might not be really important to her.
Another difference is that
This is one of the things which are really important to her.
needs no previous referent because "things" is general enough to be understood generically, whereas
This is one of the things which is really important to her.
needs a referent for "things" because "one" makes it specific:
There are many interesting things in her life, some important, some not. This is one of the things which is really important to her.
Adding a comma makes the meaning clearer:
This is one of the things, which is really important to her.
Best Answer
The type of people who work a lot.
People in English has a plural meaning, does not take an s and is, therefore, never used with a third person ending on the verb.
People like the zoo. People go fishing on Sunday. People are nice when they want to be.