We have both these items at our house, so I think I'm qualified to judge them. I hate to say it, but you're kind of at a bad age for either one of these toys. I think your daughter is too old for Leapster and too young for Gameboy. She would probably play with either one, but one would be quickly outgrown and the other would be a stretch for her.
Leapster -- Your 6 year old (especially because you say she's advanced for her age) will quickly tire of the very simple Leapster games. My 7 year old has not touched hers in well over a year. It's a fairly expensive toy. Unless you have a younger child to inherit it, I don't think you'd ever get your money's worth out of it. Also, I question its educational value. Yes, it's an introduction into the world of electronic toys /computers, and it can teach hand-eye coordination, which is a good thing in this electronic world. However, an old-fashioned book beats it out for teaching ABCs, colors, shapes, etc. My daughters never enjoyed electronic toys to LEARN those primary concepts -- they just liked making the machine's lights /sounds after the concepts were actually mastered, and I think they were in the majority. You could argue that they were reinforcing the concepts, but the truth is that they weren't focusing on the concepts -- the sounds held their attention for a while, but the Leapster was never a favorite toy in our house. If you want educational value, I say the old-fashioned methods are best for teaching preschoolers: books, blocks, crayons, and PARENTS.
Also, Leapster is larger than Gameboy and can really only be considered a car /hotel toy. Gameboy, on the other hand, can fit into a fanny pack and can be played in lines /restaurants.
Gameboy Advance -- My oldest LOVES Gameboy (she's 10), and she has particularly been "into it" since about age 8. My youngest (age 7) enjoys it as well, but she's just reaching the point that many of the games are within her scope. Many of the games are still too advanced for my 10 year old (either in difficulty level or in content -- I don't allow violent games in our house). However, I think that means that she'll continue to enjoy the toy for years, and I can continue to "upgrade" her games. Also, you can add connection cables, etc. to "upgrade" the Gameboy.
In the past, many of the games have been boy-oriented, but that seems to be changing. My youngest enjoyed Dragon Tales, but that got old in a hurry. Her favorites are SpongeBob Square Pants (Revenge of the Flying Dutchman), Groovy Girls, Barbie Pet Rescue, and the old standard: Mario Brothers. Groovy Girls is probably the easiest and I'd suggest it for a first game.
One thing I dislike about Gameboy is that you cannot rent their games; I like to "try before I buy". I suggest you read online reviews and ask friends for input before buying games. It is disappointing to drop $30 for a game that turns out to be a dud. You can buy used games for about 1/3 the price of new ones (see any game store), but the easiest kids' games don't tend to be amongst the used titles. Another thing I dislike is that the $$$ games don't come with plastic covers. You have to pay extra to protect your gaming investment.
If you're considering the Gameboy Advance, I suggest you spend a few more dollars and buy the Gameboy SP. The screen is larger and the lighting is better. We don't own the SPs (yet), but they look to be a bit sturdier. The battery covers on both our Gameboy Advances are held on with tape. Both the Advance and the SP play the same games.
Which ever you choose, be sure to invest in rechargable batteries. You'll save $$$ in the long run.
