Are there any web services that can be used to analyse data in social networks with respect to a specific research question (e.g. mentioning of certain products in social media discussions)?
Twitter's API is one of the best sources of social network data. You can extract off twitter pretty much everything you can imagine, you just need an account and a developer ID. The documentation is rather big so I will let you navigate it. https://dev.twitter.com/overview/documentation
As usual there are wrappers that make your life easier.
There are also companies who offer detailed twitter analytics and historic datasets for a fee.
Check them out!
Tweepy is one of the best libraries for analyzing and hacking around with the Twitter API. (Being a contributor for tweepy, I can vouch for it's stability and quality)
For a Python wrapper for the Facebook graph API, you can use the Facebook-Insights library, which is well-maintained and neat documentation.
There are services out there which can mine you information, but they are limited to the complexity of the query.
For example: "How many people tweeted about banana on Monday?" can be answered from existing tools
"How many people happy people tweeted about banana on Monday" can also be done, but would require more efforts and the software or tool should be able to detect emotion.
So, if you are into research, I would advise you to go with the API's and good ol' programming!