Yeah, no. All that comes to mind is the animal fat, and the fact that they used to make candles out of it (tallow dip candles, back in the 1600/1700/1800s [and maybe even farther back]). I do like candles (who am I kidding, I lurve candles), but tallow is still not a very pleasant sound. And the full tree name, tallowwood, kind of runs together in the middle.
Um, as for suggestions...do you like other naturey names? Maybe tree names? Suggestions:
Willow
Oak
Yarrow (a type of plant)
Tansy (again, a plant-sometimes considered a weed, though)
Eyebright? (Maybe too out there-it's an herb)
Meadow