When I intend to pray for someone or something, I add that intention to my general intentions. From that time on, that intention is part of my prayer intentions for all my prayers whether I conciously remember it or not, until I specifically remove it from my general intentions. There’s no special formality involved in this, I just decide to add a specific prayer intention to my general intentions, and it stays there until I decide I’m done with it.
I also pray specifically for things that come to my mind of course, and this is especially true of the Rosary, because the different meditations on the mysteries naturally bring to mind new prayer intentions as I go along. So I add them as I go along, and I also make it part of my general intentions that If I think up a new intention during the course of some prayer, that that intention retroactively becomes part of my intentions for the whole prayer.