Based on your description it sounds as if there's some kind of interference issue. Have you tried moving closer to the AP with your laptop to see if it maintains connection? (just to rule out its not some interference issue)
Is it an open access point your connecting to? (Doesn't have any encryption; requires no password to access)
The first thing I would try is a different wireless adapter (completely disabling the internal adapter). It would be preferable if you get a USB adapter with an external antenna on it. Use it for awhile, and see if it will keep a stable connection.
When your computer drops the connection from your preferred AP, it will start scanning for other APs in the area. Since you do have more than one access point within range that has the same name, the computer will automatically connect to it, whether its your preferred AP or not since they have the same name.