Compare to Solanum viarum, aka soda apple. Has a pretty extensive root system once established so digging out the complete root is advised. Otherwise, treat with a topical herbicide like glyphosate, avoiding the surrounding lawn.

Not sure what it is exactly, but it is ugly. I think I'd just dig around it to expose the root system, and then poison it. I would remove as much dirt as possible from the main root area, score/shave the root with a knife in multiple spots, and paint some Round-Up (Glyphosate) with a basting brush all over the wounds, root, and any exposed leafy part of the plant. There may be some damage to the Bermuda if you are not careful with the Round-Up (best put a towel down around where you are working).

Let the poison do its job for a week or so and spread through the plant, and it will be gone. I have killed many tree stumps this way.

Gardengal48 - I looked up Solanum viarum. The leafy parts are similar, having thorns. But I wonder if I should be seeing any of the "fruit" growth yet (or if I'm just pulling them before they reach that level of maturity). Another point of note, the root system of Solanum viarum appears to be a (what I would call) ball root system. I just dug one up that had a singular root stem with about 6 branches (straight up, none were branching out) that was about 10 inches deep. I should mention that the root snapped off at ten inches, there was more deeper. But being I am in north Texas, we have this black clay about 3 inches below the surface. Making the act of pulling any root... tedious to say the least.

In any regard, I will pick up some Glyphosate and treat the affected areas. I'm not worried so much about it killing the Bermuda in those locations (as it is Bermuda and grows back in fairly quickly). However, I wonder if I should be concerned about run off. Since I water my yard about 3 times a week - should I hold off on watering until this nuisance is eradicated?

No. Water about 24 hours after application to make the plant start a growing cycle and ensure it pulls the chemical into its system. This will speed death.

If you miss some roots when digging out a weed, pour boiling water in the hole. Never had any weed live through that. Virtually free, won't harm anything else but maybe a little of the grass around the edge.

Thank you for everyone's input. Since pulling them the other day, they haven't sprouted back up yet, so I haven't had the chance to try the above recommendations. but believe me I will!