Yes, but not necessarily in the way you think.
When most of my friends ask this question, they are really asking "Is there a creator or an intelligent designer?" That's when I grab my bag of popcorn, yell "Let the games begin!" and watch the ensuing fireworks of debate between theists and atheists. Everyone notes our seemingly cosmically insignificant planet and sun, the vast lack of undiscovered life, and the improbability of the finely tuned universe. One side interprets it as evidence for a designer; another interprets it as great chance; another, through the multiverse theory, says it's simply obvious that any universe, in which life can observe it, must be one perfectly suited to support that life. All sides understand the generally accepted Big Bang theory. One side interprets it as evidence of a designer and thereby purpose; the other interprets it as evidence that a designer is unnecessary.
But I wonder if there is another way to read the question. I wonder if it's possible to consider the question apart from the existence of an intelligent designer. Even without such a designer, can the universe have a purpose? And for those of us who do not believe in a designer, does this universe have a purpose?
The response to the question depends a bit on how you define "universe." If "universe" means the entire universe and every section, it's a much harder question: you must find purpose in every single part of it. However, if any part of the universe having a purpose counts, then it's simply a question of finding purpose anywhere. That's how I interpret the question.
But my response to the question also depends on what "purpose" means. The reason the question serves as a proxy for the hidden question "Is there an intelligent designer" is because people usually define "purpose" as "intent." If purpose is intent and if there is intent, then who intended it? We're back at the question of a designer. However, there is another kind of purpose lurking throughout the universe.
This purpose still equates to meaning, but instead of meaning being in the intention or creation of a situation or circumstance, rather this meaning is infused into a situation or circumstance through the experience itself. We experience this in the aftermath of human tragedy. We sometimes ask questions of intent. Why did this happen? What's the point of this tragedy? Rarely do we have an answer to such a question. Instead, part of moving on is not to ask why the tragedy happened, but rather to ask how can we use this regrettable, unwanted situation for good. What can we learn from it? How can we make good come out of it? How do we ensure this does not happen again?
In that subtle reversal, we infuse meaning and, may I say, purpose into a situation through our response. We may disagree whether it was designed or caused with some intention. But we know we can intentionally give meaning to a situation in the aftermath through our response. John Haught defines purpose as "the bringing about of something undeniably and permanently good." The nice aspect of this definition is that this type of purpose can definitely be done intentionally in the creation of something. More importantly, it can come about in the aftermath of an event through our response and faithful action towards giving that event meaning and purpose.
When I think of it that way, my response is yes, the universe does have a purpose. It has a purpose when people turn tragic wars into a means of reconciliation between people groups. It has a purpose when scientists, filled with exploratory spirits, give the mystery of our universe a purpose simply by uncovering that mystery. It has a purpose after natural disasters when people learn to value what is most important and to improve disaster preparedness to save lives in the future.
One of the reasons I write about faith and science instead of religion and science is that some people have a religion, some do not. But everyone has a faith--the way each lives out life based on a set of values, whether derived from a religion, an ethical system, or somewhere else. When we begin to ask questions that are not based on having a religion but on the way we faithfully live out our lives, we begin to take a step on common ground.
Maybe whether the universe has an intentional purpose is not the most important question. Maybe what is most important is whether we can imbue purpose into the universe we have. When we ask that question, we stand on common ground and take steps for the common good.