The phrase deepest hour usually refers to the time of our most dire need, despair, gloom, or despair, although it can have a more positive spin (such as the deepest hour of our relationship).
The phrase is found several times in books, often specifically referring to the deepest hour of need, or the darkest hour of night. You're welcome to peruse the several examples shown there, but I'll list my favorite, from a book called Chance published in 1921:
The night was getting on to what I may call its deepest hour, the hour most favourable to evil purposes of men's hate, despair or greed...
As for what it means in the song, as FumbleFingers indicated, we don't usually like to speculate what a songwriter means – for some songs, that can be debated for decades. However, this one isn't very enigmatic; it seems safe to surmise that song is simply saying, "You're there for me when I need you most."