Defining what it means to come up with new ideas is a very challenging philosophical question. Both humans and computers usually rely heavily on previous ideas, so it is difficult to decide whether an idea is actually new or is simply a repackaging of previous ideas. But here are a few areas in which computers have shown some level of creativity.
I'm not in any way an expert. The science section is taken from an essay I wrote for a philosophy of science class; everything else is just based on articles I've read in the past or on Google searches. (Ask yourself whether I'm being creative in writing this answer!)
Various programs have been developed that attempt to perform scientific research. Some of these simply take in data and then search for an equation to fit the data. But a more interesting example was KEKADA. The program is fed information about chemicals and various chemical reactions. It then considers possible tasks, chooses a hypothesis to test, and designs an experiment, along with an expectation of what the result of the experiment should be.
A user then enters what the results of the experiment were. Upon receiving these results, the program modifies its level of confidence in the affected hypotheses and generates new tasks to consider. After careful tuning of the heuristics involved, the program was able to trace Krebs' discovery of the urea cycle.
Computers have occasionally contributed to composing music. You can read about one such program here: A computer program is writing great, original works of classical music. The composer finished an opera by writing software to generate ideas that he could pick from.
In addition, computers have been able to study songs to identify which ones are statistically likely to become popular:
Automated theorem provers have existed almost as long as computers have. Recently, besides rediscovering well-known theorems, they have had some success at helping mathematicians discover new theorems:
Many games use algorithmically generated levels. Wikipedia lists some examples here:
Computer programs have been written to generate jokes. After some experimentation, it was found that one such program's jokes were funny about half as often as jokes written by humans:
Computers have shown some potential for creativity. But in every one of these cases, they benefit from having human supervision throughout the creative process, rather than working independently. Computers are unable to distinguish between which of their ideas are good and which ones are not.