You said you tried changing wifi channels... you might dbl-check if you are on the best channel by using InSSIDer, a free wifi network scanner. It will show you all wifi networks in the area and suggest the best channel.
Assuming you have not added a cordless phone in the last 2 weeks or any other wireless device that might be interfering, and it worked fine for 2 weeks and now is being flakey, and the channel isn't a problem....
You can get a USB wifi adapter cheap (e.g. Edimax or TP-Link, both for about $8 on Amazon and maybe less on eBay). They are micro adapters so they don't stick out. They are a good solution if you don't feel comfortable opening the machine. Or if you think the problem is temporary -- some kind of glitch that will eventually work itself out. But if you want to replace the wifi card inside the machine, that's a cleaner solution.
My wifi adapter for my Acer laptop cost like $11 off eBay. They aren't expensive if purchased used, pulled from another machine of the same model. Just choose an eBay vendor with a high rating who guarantees they have tested the part.
However you did not say if you tried connecting the laptop with Ethernet to make sure the connection still doesn't drop out. If Ethernet works fine, then it has to be the wifi card in the laptop since the other computer is also connected via wifi and not having a problem (which would tend to eliminate the router). But it doesn't necessarily mean the card went bad.... it could be something funky going on with your ports. (Have you tested all other ports on the machine? e.g. PC Card, USB, flash card ports?)
But before I ordered a new wifi card I'd Google the exact model of the HP laptop (not necessarily the wifi card model, but the laptop model)... along with 'intermittent wifi' or similar.... sometimes laptops have eccentric problems with their particular hardware configurations that others have already worked out.
EDIT: Since the router is new to you.... you say the other computer has no problems with the wifi... is it using the same protocol? For example, is the router broadcasting in g only, n only, or g and n; and are both the computers using g -- or both using n? Or is the computer that is staying connected using a different protocol?