The type of computer and graphics capabilities depends on the nature of the games. Playing games like the Sims requires more system resources than graphics enhancements, although a graphics card with a dedicated memory slot of at least 512MB memory would be beneficial.
Computers now are doing so much more so getting a laptop that wont be outdated in 3 months is nearly impossible, as technology is always changing. Now getting a laptop that will last 2-3 years is a completely different story.
She sounds like a younger child, so I am assuming that she does many things besides games on the computer, such as instant messaging, myspace, facebook, picture and music downloads, etc. These are also things to consider when purchasing a laptop. If she is like the rest of society, she is multitasking and trying to do all of them at the same time. I would like to at minimum a dual core processor. The Intel Dual Core Processors are good and keep the laptop from overheating, as was the issue with Pentium 4's which a few years back were the faster processors for gaming, but terrible at remaining cool and were large and heavy to boot as well.
Look for no less than 2GB of RAM for the laptop. The newer computers, especially Vista machines, will require more resources to run. If you can get out of it, I would advise not getting Vista. If you are stuck getting Vista, look for a 3 - 4GB RAM laptop. They should last the next couple years in resources, and require less of updating in the near future.
Hardrives are pretty large nowadays, I do not see this to be an issue, unless she downloads movies, then I might suggest an external harddrive for her to store movies.
Overall, the brand names are not nearly as important as what is inside the computer. Working in the computer industry, I have seen issues and problems with all brands, including Alienware and Sony. I might also suggest getting it at a place where you can get an extended service plan on it, especially since she is younger. Best Buy offers accidental coverage on the laptops as well as it covers the normal wear and tear on the computer, should anything happen to it within the next 3 years. I hope nothing ever happens, but I speak from experience that within the first 12 months of owning a laptop, I had to take mine in twice for repairs. Its just the nature of the game anymore. Planned obsolescence, they dont make things like they used to. Spend a little extra money on the plans to make sure you are covered.