Even cheap laptops now commonly have 160Gb hard drives. Assuming your photos are 5Mb each, at 200 photos per Gb, such a laptop could store around 30,000 photos, allowing for the operating system space.
I would recommend you give consideration as to how you will connect the laptop to the internet. Unless you intend to run additional cables, I would recommend you get yourself a wireless router and ensure that the laptop has wifi connectivity built in. That way you can use the laptop to surf from anywhere in the house.