Any of the microcomputers are perfect for hosting a web app from home, you don't even need a static IP, you can use some kind of dynamic DNS. One thing to keep

Any of the microcomputers are perfect for hosting a web app from home, you don't even need a static IP, you can use some kind of dynamic DNS. One thing to keep in mind for a write-heavy application: it will fry an SD card, so you should use an SSD attached to it instead.