Riverrun

At the end of the challenge phase, if you have not lost a challenge this round, each House Tully character you control claims 1 power.

**********************

Old Nan

Any Phase: Kneel Old Nan to choose a character. Until the end of the phase that character gains or loses a Trait of your choice.

**********************

So I've given the House Tully trait to Rob Stark during the Challenges phase. I don't lose any challenges and I have Riverrun in play. Does Rob Stark get a power token from Riverrun or not ?

From the FAQ, p.8:

(2.6) End of the Phase

Each end-of-the-phase Framework Action Window should be played without step two (Save/Cancel) and without step five (Responses). No triggered effects can be played after the initiation of the end of the phase.

“Until the end of the phase” lasting effects expire in step 3 of the “End of Phase” framework action window.

“At the end of the phase” passive effects initiate in step 4 of the “End of Phase” framework action window.

So Old Nan's "until the end of the phase" trait manipulation wears off before Riverrun's "at the end of the phase" trait-based power claim. In your example, Robb does not get power.