I'm thinking of starting a business offering items (T-shirts, bumper
stickers, etc) bearing short phrases, each of which would include the
name of someone famous. Along the lines of "I really like X," where X
could be Homer [as in Simpson], Jesus, Ghandi, Elvis, Darwin, Cartman
(from South Park), etc. I'm also thinking of including a small image
to make it clear which person we're talking about. [i.e. a donut for
Homer Simpson, a guitar for Elvis, and so on.)

I'm wondering if I would run into legal trouble using the names of
famous people which might be trademarked. Can you trademark "Homer" or
"Elvis"? Does this business idea fall under "fair use"? If not, great.
If yes, in certain cases, how would you find out when it's OK and when
it's not? If yes, in all cases, how would you get the necessary
permissions?

Some celebrities' names are trademarked. For instance, Marilyn Monroe:

http://marilynmonroe.com/business/overview.html

Someone more expert (and an official Google Answers researcher) can
come along to provide a better answer, but let me just comment that:

(1) As pinkfreud-ga points out, yes, sometimes celebrities can obtain
trademarks over their names, likenesses, and (sometimes) other aspects
of their personae.

(2) You ask how you would go about getting the necessary permissions
to use such a trademark. Basically, the answer is nothing less and
nothing more than contacting the trademark holder (or its designated
representative: agent, estate, etc.) and negotiate a deal for your
usage.

(3) Setting aside trademark, it's also true that the law recognizes
the concept of a "Right of Publicity" and a "Right of Privacy". 
Exactly how these laws apply vary from state to state, but given the
description of your project, it seems that you might run up against of
"Right of Publicity" or "Right of Privacy" laws, even if there's no
formal trademark involved.

(4) You ask about "fair use" of trademarks. One thing to remember is
that although "fair use" certainly exists, the only way to be certain
that fair use applies in any given case is to be sued, to go to court,
to argue that your use should count as "fair use", and to have the
court agree with you.  Whether or not you're comfortable with using
"fair use" as a defense if you ever get sued is something only you can
decide for yourself.

That said, when it comes to trademarks, one of the basic tests is
whether or not a usage will create confusion in the marketplace.  If a
"normal" person is reasonably likely to conclude that because you used
a trademark, then your usage is an authorized one when it is not, then
that's likely to be deemed an impermissible appropriation of a
trademark.


I am no lawyer, so take my comments only for what they are worth. But
given the kind of projecct you describe, I think you might want to
think of a broader set of legal concepts than just trademark.

I should add that this would only be using one part of any two-part
names, i.e. just Marilyn instead of Marilyn Monroe, Elvis instead of
Elvis Presley, etc.

If you use the name "Elvis" with a picture of a guitar, or "Homer"
with a picture of a donut, as you mentioned in your question, you may
run into legal difficulties, since you will be making an obvious
reference to a certain person whose name was Elvis and a certain
cartoon character whose name is Homer.

I'd also note that some celebrities maintain trademarks over different
combinations of their names.  For example, I'm fairly certain Elvis
Presley Enterprises does maintain trademarks in the name of "Elvis
Presley" and also in the name "Elvis"  So just using a single name
doesn't mean you're cleverly avoiding any trouble.

Again, I'd note that in the eyes of the law, one basic test of a usage
would be whether or not a member of the public would be reasonably
confused as to the "official-ness" of the usage.

If some guy out there saw your T-shirt with the word "Elvis" and a
pciture of a guitar, would he reasonably think that that T-shirt was
official and authorized by the Elvis Presley estate?  If yes, then
you've probably got a bit of a problem on your hands...

In the 1960s there was some legal wrangling over a rhinestoned
tee-shirt that said "Marlene" with a picture of a top hat. As I
recall, the courts found that it was an obvious reference to Marlene
Dietrich, and the shirt's manufacturers had to pay a large sum to the
Dietrich estate, which retained rights to the licensing of Marlene
Dietrich merchandise.