word choice - "Is there a" vs "is there any"?

Which one of these is right?

Is there a way [...]?

Is there any way [...]?

May I use either? If yes, what's the difference?

Answer

The expression a way alludes to one particular way. The expression any way alludes to one of any number of possible ways.

As you mentioned in your astute comment, at the time of asking the question, there's really no difference. Yes, you can use either one of them – in this context.

Consider:

Is there a way to roll babookamitzes uphill, without breaking into a sweat?
Is there any way to roll babookamitzes uphill, without breaking into a sweat?

I would say that these questions have no difference in meaning. Why? Let's assume I know three ways to accomplish this task. No matter which question you asked me, my answer would be the same:

Yes, I know at least three ways to do that.

I would not be more or less inclined to mention only one method if you used a way instead of any way, or vice-versa.

In other contexts, though, the words a and any are not interchangeable. In fact, in some contexts, replacing one with the other would result in an ungrammatical statement:

I know a way we can get to Timbuktu from Constantinople. okay
I know any way we can get to Timbuktu from Constantinople. incorrect

Any time we try to plan a picnic, it rains. okay
A time we try to plan a picnic, it rains. incorrect

But, in the context of asking a question about solving a problem, either one is just fine:

Do you know a way we can get to Timbuktu from Constantinople?
Do you know any way we can get to Timbuktu from Constantinople?
_{(answer to both questions: Yes: we can fly, or take a train.)}