Sweet spot - I think TI hit the tinkerer sweet spot with the BeagleBone, which was smaller and less powerful than its predecessor, the Beagle Board. The open nature of the board itself adds considerable value as it can be used as a reference design.
Ease of modification - For my project, we just replaced the HDMI and audio connectors with an extra Ethernet and a few micro-USB connectors and rolled out our own custom board... all in the span of a few months.
Ease of use - They need some work here for the average tinkerer who might not be familiar with the Armstrong Linux distribution. However, that just worked perfectly for me. For anyone interested, the Open Embedded Core or the Yocto project would be better resources to look at. Debian and Arch have Beaglebone focused distributions too. I would recommend against developing code on the Beaglebone itself.
So for someone looking to create a customized "device" of any kind, this would be a great choice. Also, I sell myself out by the hour doing this kind of stuff :). I would be happy to help/consult with anyone if interested.