This is of course connected to the greater question about whether anyone should lessen human suffering in the first place. I will assume ethics that would answer in the affirmative to this question, and consider whether philosophy should be part of this lessening of suffering.

The problem is that it comes down to "what is philosophy?". The question is huge, and I have often thought that this would be the last philosophical problem ever solved, if they indeed can be solved. But we do not need to choose a definition for philosophy, luckily, to consider the question you posed; we need only to see that all plausible candidates for an answer share in common the feature, or not.

What if we started with considering why we do philosophy at all? Proteus already offered Hegel's idea, which I consider a rather good one. Kant would say that doing philosophy is ultimately about showing that man is free and to put up an ethical system. Simply: to answer "what should we do?" (his theoretical philosophy is subsumed under practical, but it also answers a question: "what can I know?"). Levinas says in one interesting passage that "Politics [the art of foreseeing war and winning it by every means] is opposed to morality, as philosophy is opposed to naïveté", which shows that for him philosophy is about opposing naïveté: not about finding different results than we already have, but about giving them a thorough examination - in a sense, about justifying the common sense views (of course it may end up refuting them, but that is not the purpose of philosophy - the same line of thought pertains to science as well). Then philosophy for Levinas is more or less about clarity and solid foundations. Wittgenstein thought philosophy was therapy, Heidegger started from being in the world towards understanding this world. The Scholastics wanted to make sense of the world God had created for us - to make sense of a world given. The Ancients shared some common features of philosophy as well, mostly directed at good life, happiness, the Good, etc. Descartes wanted to get to the bottom of things much like Levinas: through careful and relentless examination of our foundations for what we seem to take for granted.

What is then philosophy, according to these views, about? I think it is fairly safe to say: about making sense of the world we inhabit, and of ourselves. It takes the world that is so chaotic and labours to make a sort of picture of it. In the process we have sciences that tell us what the world is like and what laws it obeys, logic that tells us how we think and how concepts hang together, ethics about what we should do to each other and what not, politics about what the society should do, and how its people should be governed, etc etc.

Now I think the question very general, and I think it can be easily answered: if philosophy is about clarity, about "making sense", then we need only to consider whether having muddled views, things not making sense etc make us suffer. I think it does: we do suffer from the condition of being uncertain, about not knowing what something is about. It makes us lose the sense of being free, being in control. It makes us frightened, shaky. So, in that sense, it is not only about philosophy "accidentally" lessening suffering, but about it essentially doing that. Philosophy is about lessening one aspect of suffering.

The second portion of the question is also answered in this: by, as Cuthbert put it, promoting clarity. Each and every connection we make in our minds produce satisfaction, and not only for ourselves, but for others as well. It is not only the philosopher that finds satisfaction in his philosophy, but also others that hear about it. But there is of course the question: what if philosophical considerations lead to suffering? That is, if I end up thinking that I am the only person that matters and others are for me to "harvest". I am not sure if this can be answered, at least not in short, because it is about ethics in general. What is the correct ethical view? Who knows. But what I do think I know is an interesting thing: there are no philosophers that would advocate a view that should lead to more suffering. Even such philosophers as Nietzsche had the ultimate goal of freeing man, to make him better. To overcome nihilism. We do have a lot of people who end up thinking in those lines, but I would simply ask: did they do that because of philosophy, or did they think egoistically before and then only tried to find justification for that? If so, would it not again be philosophy that would tear down their false justifications and show their errors? I tend to think so, for I have never seen a view that would promote the suffering of people and would stand in the face of a more careful scrutiny.

So, as an ending remark I think philosophy should contribute to the lessening of human suffering regardless of the view of philosophy, because philosophy is always necessarily about clarity, about "getting to the bottom of things", and this lessens suffering. It may not do it in the same sense the Red Cross does - it does not give food nor cure diseases - but it does it nonetheless. This is a strong claim in many ways. For one it is based on saying "all philosophical views share this in common" and it can be brought down with one diverging view. It is also a problematic view, because it is based on saying "all diverging views can be shown not to be philosophy", so it goes around in circles. But in the end it must, because it must be a definition of philosophy we work on - and so by definition things that do not fit it are not philosophy. But I would also agree with Rhodus in that we should not say what the "purpose" of philosophy is. And I am not saying that either: I am saying, basically, that philosophy cannot help but lessen the human suffering. Individuals primarily, groups and societys with some luck. The purpose of philosophy may be different for different philosophers (is not philosophy defined through what philosophers do?), but whatever it is, the aspect of lessening suffering is always manifest.