For years, my 3 eldest kids all attended 2 week day camps with one overnight and loved it (for a few years, they got to do every session because their Mom was camp nurse those years). All of them went on to be junior counsellors. The 2 youngest (10 and
have done it now for the past two years, but this summer was the first time that they elected to do the optional 1 overnight. Conor (10) was ok with it, Caelin - just turned 8 at the time - was in tears at the last minute, but stayed and enjoyed it. As a matter of fact, they then went on to a 1 week overnight camp and both enjoyed themselves thoroughly (no tears at all).
I only know Camp Nazareth, of which David wrote, from reputation, but that reputation is excellent, as he's said. If there is at least one other child attending whom your daughter knows, I think it helps tremendously, particularly if she's a bit on the shy side. I can sympathize with Dad's hesitancy - I had the same with each of mine the first time, especially my little princesses, but they all loved it, thankfully. (And, yes, there are kids who won't - not at all sure that I would have at the age, but I never had that opportunity, so I'll never know.)