Well, League of Legends is not such a heavy game. It needs at least 3Ghz CPU and 4GB RAM are recommended so any laptop at the price of ~550-600$ I think would be ok. But you have to look at the graphics cards, it needs something like NVIDIA 8800, but I can't really recommend anything since there is a big difference between ATI and NVIDIA and much more between Desktop (8800 is a desktop GPU) and laptop. It needs at least 512 MB VRAM (the GPU memory) but most of the new laptops have 1GB so this is not going to be a problem.
However I would recommend a laptop with ONLY dedicated GPU. You may want to have a laptop without integrated graphics (a bit difficult to find because most of them (mostly with AMD Fusion (A6-A8-A10)) have integrated graphics) because there is a possibility that your laptop will use only the integrated graphics in the game and you won't get good graphics performance because the dedicated card will just sit there doing nothing if it's not configured correctly. I have a new laptop with A6 and Radeon HD 6470m and I have this problem, so you might want to check on this before you buy a laptop.
As for some brands you asked, I don't recommend HP, they might look good but after a while, problems will start coming up especially with their battery. ASUS makes some good notebooks and they obviously will have good GPUs. I have a Samsung notebook, it's pretty good, I upgraded to a SSD and it's really fast. If only I didn't have this problem with the graphics cards (the one with integrated and dedicated GPUs)... But their Costumer Support really dissapointed me so I have to figure it out by myself... I don't recommend a Sony laptop. You will pay too much and get much less than you will expect...
I hope I helped...