A trick to change the artboard size to be centered:
In Illustrator the functionality is there.
Actually, since its beginning the resizing of the artboard was only on the right side! The new CC versions permit you to resize from any side you want, including the center.
There's a few ways to achieve what you want:
Create an empty rectangle or square that is centered with your artwork.
Then select that shape, go in the menu "Object" > "Artboards", then select "fit to selected Art."
You can also simply go in the document setup, select "edit artboard" and resize the artboard by holding the command + option (alt) + shift key; it will resize the artboard equally on each side. Holding these keys together work as well to resize a lot of things in this way in pretty much all Adobe's software.
Finally, you can also go in the document setup, then "edit artboard" and manually enter the new dimensions on the top menu bar and put the "reference point" in the middle. It's the most precise way to do it. If you want to resize from left/right/top/middle instead, simply change that reference point.
Unfortunately, the "command + option(alt) + shift" doesn't work on Photoshop artboard if you resize manually.
The only way you can do it is the good old lazy way, using guides, unfortunately; the same way as you maybe do sometimes when doing free transforms. I'm quite sure the option to do it automatically will be added on one of the next Photoshop versions.
Make your rulers visible (R) and then use the "Artboard tool" to see the handles on the corners and the ones at the center of your artboard. By default, you artboard is already centered.
Then add a vertical and an horizontal guide aligned with the middle handles... then resize manually your artboard or use the measurements in the menu bar. And then adjust your new artboard using these guides as reference, simply by aligning again the middle handles with the 2 guides.
Really not the perfect way but still faster then trying to measure the XY position for each situation.
Adobe CC 2015 released June 2015 (for paid subscribers):