I have solved this one. It is a Rockettheme issue, as suspected. The more recent Rockettheme templates have custom input styling, which is the problem. In these templates, in the 'advanced' parameters, there is the option to add parts of your site that must be excluded from this custom input styling. To that list, I added 'com_alpharecommend'.
This did the trick. Now the OI 'address book' opens in a beautiful popup and imports as it should.
Except, it does not import from Facebook, Twitter and somewhere else, which I find a little odd. I have used other OI components that did import from Facebook.