Note: you will need to setup SSH to follow the instructions below.
The splash screens the reMarkable uses are stored as PNG files in /usr/share/remarkable
root@remarkable:~# ls -la /usr/share/remarkable drwxr-xr-x 4 root root 1024 Jan 3 13:24 . drwxr-xr-x 23 root root 1024 Jan 3 13:24 .. -rw-r--r-- 1 root root 65309 Nov 9 15:59 batteryempty.png -rw-r--r-- 1 root root 75499 Nov 9 15:59 poweroff.png -rw-r--r-- 1 root root 63863 Nov 9 15:59 rebooting.png -rw-r--r-- 1 root root 63638 Nov 9 15:59 restart-crashed.png -rw-r--r-- 1 root root 52644 Nov 9 15:59 sleeping.png -rw-r--r-- 1 root root 2629366 Jan 3 14:22 splash.bmp -rw-r--r-- 1 root root 59839 Nov 9 15:59 starting.png -rw-r--r-- 1 root root 63586 Nov 9 15:59 suspended.png drwxr-xr-x 2 root root 2048 Jan 3 13:23 templates -rw-r--r-- 1 root root 197 Jan 3 13:22 update.conf drwxr-xr-x 3 root root 1024 Nov 9 15:55 webui
These files can be copied off the reMarkable with scp over ssh and then edited in a normal image editor that understands PNGs and copied back. Note that transparency, if present, is taken into account by the tablet (meaning that the former content remains visible where the PNG is transparent; this is used e.g. by sleeping.png in the 'light sleep' mode).
The filenames are reasonably self-explanatory, but it is worth noting that a second copy of splash.bmp is stored in /var/lib/uboot/ (where a FAT partition for booting is mounted): you have to copy it over there if you want it to be shown when you power up the reMarkable. It also expects a specific version of bitmap format, specifically version 4, so you need the file utility to yield PC bitmap, Windows 95/NT4 and newer format on whatever you replace it with. One way to achieve this is to edit your image using GIMP and enable greyscale mode before exporting to .bmp, which will then set it to version 4 automatically, instead of version 5 if you had maintained colour, which cannot be displayed by the bootloader. This image should also be rotated 90 degrees clockwise.
Also worth noting is that the reMarkable displays a progress sequence of dots in the middle while showing starting.png, so it might be worth adding your contact details (or other changes) where they won't be obscured by this graphic.