PC

Several open hardware project that are building something with most of the functionality of a standard desktop PC:

ARMUS Embedded Linux Board
Linuxstamp
the Kestrel
Balloon Xscale ARM+FPGA dev board
Computer Components wiki: Custom Notebook
the OpenBook project is an open source hardware and software project designing a computer tablet ... Everyone can contribute and help to shape the OpenBook hardware and software specifications on the OpenBook Project website. See the Openbook wiki (http://obook.info/)
wikibooks: build a laptop has a few tips on a semi-custom notebook computer (but it doesn't have enough flexibility to, say, make one with a RAID1 mirroring).

Boards that don't run Linux (yet), including

Many of the "TTL CPUs" listed in | Wikibooks: Microprocessor Design are more-or-less open hardware. Several of them have video output and keyboard input connectors.
Several of the Parallax Propeller Development Boards are more-or-less open hardware. Many of them have VGA output and keyboard and mouse input connectors.
"The Maximite is a small and versatile computer running a full featured BASIC interpreter with 128K of working memory. It will work with a standard VGA monitor and PC compatible keyboard and because the Maximite has its own built in SD memory card and BASIC language you need nothing more to start writing and running BASIC programs. The design is free and open source including the software and BASIC interpreter."[1][2][3] The "haiqu" has ported BSD to the Maximite and UBW32.[4] Apparently (?) also supports Arduino shields.
"Special Computing" has a list of SBCs and accessories -- BeagleBoard, IGEPv2, HawkBoard, etc. -- Can any of them run Linux?

Many of the other open circuits projects are designed to plug into a standard desktop or laptop PC.

open laptop

Bunnie is building Novena, an open laptop motherboard.

"Building my Own Laptop": "all the design documentation is open, so others of sufficient skill and resources can also build it. The hardware and its sub-components are picked so as to make this the most practically open hardware laptop I could create using state of the art technology. You can download, without NDA, the datasheets for all the components, and key peripheral options are available so it’s possible to build a complete firmware from source with no opaque blobs."
"Update on Our Laptop (aka Novena)": "All our progress has been publicly trackable via our git repos and on our wiki. There’s also a discussion forum"
"Posts Tagged 'novena'".

How to test a notebook computer? Does TestTheHardware already include those tests?