As we learn of more and more contacts between members of Trump's team and Russian officials or go-betweens, both in the United States and in Moscow, let's engage here in a few theoretical questions as to what might have been going on. And we don't need to get too conspiratorial about this; at this point, the simplest explanations are still likely the best.
And the simplest explanation for Russia's peculiar attentions towards Donald Trump's top advisers is the one we already know. From the declassified version of the intelligence community's report on Russian state espionage efforts during the 2016 election:
Russia’s goals were to undermine public faith in the US democratic process, denigrate Secretary Clinton, and harm her electability and potential presidency. We further assess Putin and the Russian Government developed a clear preference for President-elect Trump. We have high confidence in these judgments. [...]
We assess Putin, his advisers, and the Russian Government developed a clear preference for President-elect Trump over Secretary Clinton.
This "preference" for Trump developed late in the campaign. The original goal of Russian hacking was, according to intelligence efforts, similar to previous Russian efforts in other nations; to sow mistrust of democratic institutions in general and of Russia-damaging news stories in specific. That goal morphed into an effort to support Trump specifically, in the closing months—after, as we're now learning, numerous meetings between Russian government figures or proxies and member of the Donald Trump campaign.