You could use the "American Dream" as a theme for a Beating the Odds speech.  There are any number of good stories of someone in history who went from rags to riches simply through hard work and perseverance.  Bill Gates and Andrew Carnegie are a couple of examples, but I think the stories of immigrants who have overcome all of the barriers to success in the US are more compelling.  Colin Powell or Madeleine Albright if you want famous people, but there are undoubtedly examples of people you know who fit that definition as well.

Find examples of schools that are high-performing despite the fact that their demographics say they shouldn't be. You could also talk about the career of any professional athlete, because the odds of anyone becoming a professional are astronomical. You could talk about someone winning the lottery. I like the idea posed earlier about the (now deceased) octopus that predicted the World Cup results. You could deal with it on many different levels, whether you're looking for an inspiring story or simply a story about someone doing something that is simply mathematically anomolous.

Post #4's comments remind me of the history of Lake Erie which became so polluted that ecologists thought that there was no hope of saving this Great Lake. However, Lake Erie "beat the odds" as it is today a lake that yet supports life.

You may wish to read about the turnaround Lake Erie made at the following site:

http://www.time.com/time/magazine/article/0,9171,948661,00.html

When I read this, I thought of cancer.  Surviving cancer is often considered “beating the odds,” so why not choose examples where people seem to not be able to make it but do.  I would consider researching the odds of surviving cancer (maybe choosing a specific kind) and then describing the story of a person who did. 

Another example might be a foster child who was successful as an adult.  Most foster children do not grow up to be runaway success stories.  You could describe the statistics and then focus on the story of one person.

"Beating the odds" to me is the story of someone who has accomplished something that no one thought they could accomplish.

There are probably plenty of real life stories you could include in this theme, so I would caution you to perhaps come up with a list of those that are the most overdone.  For starters (not to diminish the stories themselves, but to avoid sounding cliche):

1. underdog teams winning the championship
2. overcoming illness, disease, or injury

Have you ever "beaten the odds" in your personal life? That would be an ideal anecdote to start your presentation. In the absense of a personal story, finding the story of someone else who has beaten the odds would work.

You might consider giving examples of approaches used by persons who have "beat the odds" to keep themselves motivated and focused on the tasks needed to reach their goal.

"Beating the odds" can apply to so many areas of life and environment, as has been pointed out. You can talk about how someone beat the odds by getting well. You can talk about how an ecosystem beat the odds by being regenerated,

http://www.ncrs.fs.fed.us/4101/local-resources/docs/palik.pdf

http://www.fao.org/DOCREP/ARTICLE/WFC/XII/0323-B1.HTM

You can talk about how the world hopes to and is taking measures to beat the odds by reestablishing nontoxic nutritional food and by simultaneously reestablishing humans' link to food and the food process and by creating sustainability.

Europe
http://www.slowfood.com/international/1/about-us?-session=query_session:475405030c4240F3C1xj48637800

United States
http://www.slowfoodusa.org/index.php/slow_food/  

For beating the odds, you could really use any person who has become extremely successful.  After all, the odds are that any one person will remain obscure their entire life.  You could, for example, talk about some athlete who has had to overcome a difficult childhood.  

As for the animal, what about that octopus from the zoo in Germany that predicted the results of the games at the World Cup in 2010?  That's beating the odds.

Anecdotes are always a good idea. They help the audience focus on a real situation and real people (or animals). Just make sure your anecdotes are verifiable and believable.

Since your topic is "Beating the Odds," it would be engaging if you managed to actually include some odds in your talk. That might be hard to find, but it sure would work nicely.

Your topic is very, very broad. You want to narrow it down some. Think of a specific situation in which beating the odds would be an interesting thing to talk about, such as overcoming physical disabilities, etc.