I have had similar issues with an iPod touch and having it reliably connect to other (non Apple) home networking equipment.
There are too many variables in your set up described above to diagnose an exact problem. But the types of things you want to try are:
1- Further problem isolation.
You have found that rebooting the router fixes the problem. But what about rebooting just the iPhone? Or rebooting the computer? How about if you just turn the WiFi on the iPhone off then on? Do the iPhone and the iPad leave the house during this time and connect to other networks? What happens if you leave the device in the house for a few days?
2- Depending on the outcomes of the testing above
You might want to try using fixed IP addresses for some or all of your networking components. You do this on the iPhone in Settings->WiFi->YourNetwork->Static. If you are not familiar with this sort of stuff note down the settings on the DHCP tab and copy them into the Static tab. You can always change back to DHCP if things get messed up. I use fixed IP addresses on my iPod touch to mostly solve my problem. See here: http://www.net.princeton.edu/apple-ios/ios41-allows-lease-to-expire-keeps-using-IP-address.html for highly detailed information about a DHCP bug in iOS.