One of them has left his umbrella.

or

Someone has left his umbrella.

The situation: one man of the group has left his umbrella, but we don’t know who exactly.
Which sentence is correct in this situation?

1. Someone of them has left his umbrella.

2. Some of them has left his umbrella.

As noted above, we'd typically say instead 'Someone has left his umbrella'.

#1 is wrong. However, #2 is correct but literary in tone, formal, stylish and rather old-fashioned.

The following excerpt was wriiten in 1938. It means that one of them occupied the land, but it doesn't tell us which one.

. . . he died in 1706. His three sons, George, William, and John, inherited and some one of them occupied the land.

 http://www.rootsweb.ancestry.com/~sccalhou/history/earlysettlers.htm

Published in The Proceedings of The South Carolina Historical Association, 1938

Note that you can also use the same construction for numbers greater than one, as in the following extract. 

The latest careful-what-you-wish-for park is Theodore Roosevelt National Park in western North Dakota. They received 47 elk from our Refuge in 1985 and are now wondering how to get rid of some 300 of them.

http://www.planetjh.com/news/A_105232.aspx

Published in the Jackson Hole Weekly, in 2009

Finally, although this construction is correct, it's not one that I'd normally teach to my English students.

Best wishes, Clive

1. Does the expression “some of them” or “some of those people” always mean “numbers greater than one”?-- Yes.

So the correct sentence is “Some of them have left their umbrellas.”?-- Or 'umbella'.  Yes, if there is more than one abandoned umbrella.

2. Which form (with “one” or with “body”) is used oftener?-- I'd say that '-body' appears more in speech and '-one' appears more in writing.