Some 'oddness' here. Youi said every third or fourth photo came out okay, and the remainder were OVER-exposed??? That is exactly opposite of what I would expect from a flash that was not recylcling ast enough, unless your camera was setting an ambient light exposure that was causing long shutter speeds.
You did not mention the most-critical technical elements: what camera and lens were in use? A slow lens, like an 18-55mm might have been at f/5.6; when used on a consumer-level camera, like a D5100 or something like that (Canon Rebel XX,etcx), f/5.6 is sometimes the bare minimum amout of light for the AF system to work--and that goes more than doubly so if the basement was as dark as you stated. In BRIGHT sunny conditions, many lower-end autofocus modules will focus when the lens is an effective f/5.6 aperture; but add a teleconverter, or Neutral Density filter or polarizwer, and the same identical camera might have tremendous difficulty focusing rapidly, or reliably.
Anyway...SINGLE-point autofocus in dim lighting is often not as good as multi-point AF is. Why? LESS data. ONE point of data. If that one, single point is on say, smooth skin at close range, the target is in effect, to a computer, a "low-contrast target". IN marginal conditions, low-contrast targets can be very difficult for AF systems to lock onto. However, if say you used a small number of clustered AF points, and also were aiming at a child's face from close range, a smart, modern, color-aware, distance-aware Nikon's AF system, combined with its metering system, can note the eyeballs, note the dark color, run the computer simulations stored in memory (compiled from tens of thousands of actual photographs and decades of Nikon know-how), and the AF system can get a lock with very little effort, shot after shot.
I'll say it again: using SINGLE-point AF with a slow lens, in dim light, is not a good strategy when you need to focus FAST, and reliably. Use a small cluster of AF point, with most modern Nikons.
Light: I usually use Nikon's "fifth battery" on the SB 800; it's a special door to the battery compartment. It fits in place of the normal door; adding that fifth AA-cell REALLY speeds up the flash recycle. But, when I need to shoot FAST, I like a small studio power pack, set to LOW power. I get one-second recycle times. As many shots as needed. If you want to simulate the adorable pinbk backdrop cake smash shot with a flash setup, you need a large modifier that "mimics" that kind of light. If it is LARGE, and set up at 10 feet, then there is almost no falloff from lkeft to right on the set, so the kid can move or crawl, and the f/stop will stay pretty constant across the posing area. MY suggestion? A LARGE 72x72 inch, white panel with two studio flash heads aimed through it, and another similar panel, or a wall, on the opposite side. If you do not have a panel, use a white wall to bounce the flashes off of. Use wide-angle reflectors to give a good bounce coverage, and make sure to use a good lens hood, and better yet, block off the lens with a go-between sheet of foam-board or other device. What you are looking for is a "north light studio" effect.
But yes, they were completely underexposed and very inconsistent. I would have one image slightly overexposed and it would dwindle down ( way down ) from there. 
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