Micah Meyer, horticultural ag agent for Jefferson County, wrote in to say he thinks the flower Judy of Lumberton has been trying to identify is neither a Peruvian lily, as I thought, or a walking iris, as PJ thought.
Micah said he can't tell for sure, but the flower looks like what is commonly called "blue-eyed grass" from the genus Sisyrinchium spp. One of his plant books says that there are around 16 species in Texas and that they tend to hybridize readily. That could explain why it isn't an exact match when I look it up, though it looks close. The only difference is the leaves of blue-eye grass are skinnier.
Dig-It! reader Margie said the plant is Aristea Ecklonii, commonly called Blue Star. The avid gardener sent along a photo of her plant - and it's about as close a match as I've seen. Here it is so you can judge for yourself - and thanks, Margie and Micah.