CrazyThoughts.com has an extensive list of silly questions and oxymorons to ask your friends. However, some of the best "random" questions are best invented by yourself.
Random questions should emphasize the humor of the absurd and the non-sequitur. They do not need to have easy answers, or even answers at all.
Random questions can touch on any topic. Drawing from popular culture is often a good way to generate random, off-the-wall questions. Oxymorons and paradoxes, although not "random" in a strict sense of the word, often can be used to create interesting questions. For instance, think about the phrase "act natural," a simple oxymoron. You could ask "How can you act natural, if it's a kind of act?"
Even philosophy can be used to generate random questions. A question that occupied philosophers in the Middle Ages was "How many angels can dance on the head of a pin?" This is not a simple question to answer. Another question from Medieval Philosophy was "Did Adam and Eve have navels?"
The following are some great random questions. Do deer have friends? Why is cake? Where is the biggest sponge cake in America located? Which type of cheese reminds you most of your parents and why? Do ghosts miss food?