Why did Doctor Who (2005) Season 7 go off air for 6 months?

Budget issues. Since 2010, BBC's budget for Doctor Who has been shrinking, causing issues for the series. One of the efforts to deal with the issue is to spread out production and airing of the show, decreasing the number of episodes made each year. This is why Series 6 and 7 have done the 'split season' thing, airing half a season, then airing the rest a few months later.

From an interview with Steven Moffat (May 2012):

There are going to be five Doctor Who episodes in the autumn, then a Christmas special, then eight more in 2013 - what was the thinking behind that structure?

• I don't know, on this occasion, that the thinking particularly came from me, actually. I've always been open to anything that shakes [the series] up. I think that decision actually came from the BBC.

  But I've been well up for anything that we can do to shake up the transmission pattern, the way we deliver it to the audience and how long we make the audience wait, simply because that makes 'Doctor Who' an event piece.

  The more 'Doctor Who' becomes a perennial, the faster it starts to die. ''You've got to shake it up, you've got to keep people on edge and wondering when it will come back.

  'Sherlock' is the prime example, as far as that goes. 'Sherlock' almost exists on starving its audience. By the time it came back this year, 'Sherlock' was like a rock star re-entering the building!

  So keeping 'Doctor Who' as an event, and never making people feel, 'Oh, it's lovely, reliable old Doctor Who - it'll be on about this time, at that time of year'. Once you start to do that, just slowly, it becomes like any much-loved ornament in your house - ultimately invisible. And I don't want that to ever be the case.