Significantly, Ned emphasises that Jon has Stark blood without saying he is his son, and we learn that Jon knows nothing (heh) about his mother. It's clear it's a sensitive subject for Ned, and he gets incredibly emotional. The "I promise" line reinforces Ned's connection to Lyanna, and the promise he made to protect Jon.
Which is exactly what he did Jon's whole life, and what he's still doing here – the reason he delays telling Jon the truth until "next time" is because Ned won't reveal anything until it's safe to do so, i.e., after Jon has joined the Night's Watch and therefore not only renounced any family titles and claims, but is also living as far from Robert's reach as he can possibly get within Westeros. Unfortunately, as we know, Ned and Jon never get to see each other again.