Laptop all running smoothly - could ping it no problem and connect to it via \\name.
But no way could I http to a website on the laptop.
Did the "ipconfig /flushdns" dance - no joy.
Double-checked all the IIS settings.
Had a chat with Mr. Google and found a suggestion to:
telnet "IP address" 80
This tests if port 80 is open.
No joy - aha - so it's not IIS per se - it's the TCP/IP traffic on port 80.
Run up Windows Firewall. Yup - HTTP traffic on WWW was disabled for port 80. Allowed that - bingo - all A-OK.