Can you refrigerate cake batter?
Yes, usually for a short time. How long do you want to refrigerate it for?
Of course you can
It depends on the batter. Some batters rely on heat to produce lift so it is important to mix, pour and get the cake in the oven as soon as possible. Other types of batter do well if allowed to rest. I don't know of any cake batters that are improved by refrigerating--if it is not a finicky recipe you might be able to refrigerate it but I suggest allowing the batter to come to room temperature before baking.
@Slow Cooked Pittsburgh there are in fact cake recipes that call for batter refrigeration, namely genoise and sponge cakes, madeleine batter. I like to rest my madeleine batter for about 6 hours before baking, already spooned into the molds. It helps the bubbles to set up and the backs of the madeleines form perfectly. Chilled batter in a hot oven also helps the cookies get a uniformly golden color without too much browning.
Chrissy, I think it's fine as long as it is less than 24 hours. If your fridge is very cold, the batter will start to harden. I'd also put a film of plastic wrap over the batter- some batters form a skin.