Upgrading to SQL Server 2012 from SQL Server 2008 R2
Benchmarks aren't done as frequently as before, at least not publicly. This is true across the industry, not just SQL Server.

Internally, performance improvements are always pursued for each major release, though it may target some specific areas or something broader. One area that still gets somewhat regular updates are TPC-H benchmarks. Check out the TPC website and filter on SQL Server. 

That said, you really should get your customer to reconsider their business preference on "upgrading" to SQL Server 2012. Mainstream support for 2012 SP3 ends in July 2017. See the Microsoft product support lifecycle search page for details.

That means they're investing in the upgrade for a product that is out of mainstream support within a year. The business might think it's ok, but make sure they understand the impact. Definitions of mainstream and extended support are documented in the Microsoft Business, Developer and Desktop Operating Systems Policy.

Fixes aside, some industries have specific requirements about only using products that are still in mainstream support so make sure your client is not affected there.

Try to get them on 2016 or at least 2014. You'd be doing your client a huge service.