Trying out a number of distributions I got the following results:
Tiny Core (1.2): Booted and ran with no problems at all.
DSL (4.4.10): Booted and ran with no problems at all.
Puppy (4.2): Booted and ran with no problems at all.
On a recent revisit I found Tiny Core 3.7 booted quite happily from a USB pen drive. I was also able to install it (using tc-install) onto an 8GB hard disk connected to the IDE interface.
However things were not the same with Tiny Core 4.0. Whilst it booted quite happily into its desktop, I found that it could lock-up. I do not know the actual cause of this but with my limited testing running tc-install is a good way of bringing this on. This implied that maybe it was brought on by accessing the IDE interface - it's not the case. Having (re)booted, I opened a terminal, used fdisk to partition the hard disk, and then used mkfs.ext2 to create a file system on it....paused to write these words.... and then the system locked up again a few seconds after I returned to try something else.
One obvious point of difference is that Tiny Core 4.0 uses the 3.0.3 kernel whilst Tiny Core 2.7 uses 2.6.33.3.
So something that still needs to be investigated....
Another point is - Is this a VIA chipset issue or something unique to the t5520 or HP?
Tiny Core has now reached version 13.1. This happily booted from an 8GB SanDisk Cruzer USB drive and subsequently was installed onto the internal DOM.