You probably do not want to use a Linux distribution intended for a PC desktop or laptop or tablet for an embedded system. For embedded devices, the Linux kernel is often built (configured and cross-compiled) from scratch, although the Buildroot script makes that task quite easy. Besides customizing the kernel to the exact requirements of the hardware and the application, you can specify non-standard schedulers and memory allocation schemes. For conservation of main and flash (storage) memory, embedded system typically use the unified BusyBox instead of assorted binaries for commands and replace the huge GNU libC with the compact uClibc.
BTW the Tizen and Meego projects seem seem to be way beyond what you need, unless I'm underestimating your design. Those projects are for interactive panels, whereas I'm guessing you are building a visual panel to display sensor data that has minimal user inputs. Isn't your project in the instrumentation category?
This will use an ARM processor
That covers a rather broad spectrum of capability. I'm guessing that you'll end up with an ARMv9 SoC. But if cost is a severe constraint, then an ARM Cortex-Xx (ARMv7 based) with a simpler RTOS (i.e. no MMU) might suffice.
This needs to have a fairly quick boot time, 5-10 seconds
Linux will require some work to achieve a short boot time (e.g. quiet boot option). But I've only read of boot (U-Boot + Linux) times lowered to the 12-15 seconds range. You should quickly decide on a tentative chip, and then obtain an evaluation board to test out some design concepts on this prototype.