"The rez cost of the first piece of ice rezzed each turn is increased by 1."
When I first read it, I read it as: "increase the first rez cost each turn by 1", but this isn't actually what it says.
There are three situations that are interesting:
a) a piece of ICE is rezzed multiple times in an encounter (e.g. TMI);
b) a piece of ICE is rezzed multiple times in a turn (e.g. Cell Portal);
c) a piece of ICE, which was the first piece rezzed in a turn, is re-rezzed for whatever reason in a future turn (e.g. Emergency Shutdown 'victim').
I think that by the wording on the card, all of these situations should have the ICE at 1 over normal rez cost as there is nothing to suggest that the rez cost increase of the appropriate ICE is temporary: it is an effect on a particular ICE determined by the trigger of it being the first to be rezzed in a turn (and a piece of ICE should even be able to be affected more than once if it is the first piece of ICE rezzed on multiple turns). Reina Roja's ability only lasts for the duration in which the Corp is paying to rez the ice.