cloud computer to speed up sql big data queries?
You've asked a question which often results in real projects (and I freely admit to being a consultant in this area).

I'm just going to give you some things to think about:


If you are getting your results from the cloud, you have to get your data there.  This may involve additional lags.
How real-time is real-time?
Do you have programming expertise in the group/company your work for?  For complex data manipulation tasks, Hadoop (and its derivatives) is a cloud-based technology that works well.
Once you've found something of interest, do you have a way to "productionize" it, so updating the results does not waste analysts' time.
The cloud (such as AWS) can provide very scalable hardware.  However, hardware is pretty cheap.  Part of the question is the balance between expenses versus capital investment.
Are you tied to MySQL?  Other databases have better functionality for complex queries on large data sets.