There is no built-in option for a member selected background image in phpBB3 and I've never seen any themes or mods which support anything like that.
The custom style we use here is based on the standard subsilver2 theme and it doesn't use a background image, only a color.
If a background image were added, it would replace the blue background color (shown below) everywhere on the website, so an image would need to fit with the color scheme of the site or it wouldn't look so good.

The link to the BG image is normally static and stored in an external CSS file but the board could certainly be modified to allow a member to specify a BG image. The CSS could be in-line and use a member's image link instead. I'm not sure how practical it would really be with this theme (floating rather than fixed width). It may be more practical with another theme or a theme that was designed with that in mind from the start.
I think would be more practical on a fixed width theme, where the BG tile image makes up the borders on each side of the forum. Changing the BG image in that case, wouldn't affect anything else on the site. It's actually a pretty cool idea for themes like that.