(Perhaps this should be merged with Paul D. Waite's answer, but I don't know how to do that.)
The way the trip to 1970 is introduced as Tony's idea in the film definitely makes it a little bit mysterious. The importance of the situation is emphasized by Scott Lang totally losing his cool and ranting about how the mission has failed, accusing Tony of not wanting it to succeed, and otherwise being of no help whatsoever.
Tony doesn't have an exact date or details, yet he seems completely certain that the two things they need (the Tesseract and Pym particles) can be found there. Steve Rogers wonders how he can be so sure, and the audience is invited to wonder the same thing.
TONY: I got it. There's another way. To retake the Tesseract and acquire new particles. A little stroll down memory lane. Military installation, Garden State.
STEVE: When were they both there?
TONY: They were there at a...I've a vaguely exact idea.
STEVE: How vague?
SCOTT: What are you talking about? Where are we going?
TONY: I know for a fact they were there...
SCOTT: Who's they? What are we doing?
TONY: And I know how I know.
I think the mystery of Tony's "vague" yet "exact" knowledge is answered when Tony meets his father in 1970 and Howard Stark mentions that his wife is expecting a child-- this is shortly before Tony Stark's birth.
Our memories of events from decades ago are cloudy. It's unlikely that Tony could have remembered when SHIELD was running experiments on the Tesseract or which times Dr. Pym was on site-- except that he had the one sure frame of reference of his birth date. It explains how he could be so certain without being able to give any details. Our memories tend to be very self-focused. (Perhaps even more so in the case of Tony Stark.) For example, I couldn't tell you what my grandfather's job title was, but I could tell you the exact date of a worker's strike at his company because it happened on my birthday and he told me about how he was afraid he was going to be stuck there and miss the birth of his grandson.
I think it's an effective storytelling move because it brings the focus away from the cosmic and fantastical and puts it in the realm of the personal and mundane. How do you defeat the big purple guy named after Death? With the simple connections with your family and past that make up this thing we call Life.