ADS Controller described event when a VFR departure aircraft failed to display a data tag and the required follow-up coordination was questionable; adding the acquiring and disappearing data tag problem was a random issue.

Narrative

I had just accepted the local position from a developmental. Aircraft X had been cleared for take off from Runway 15 by the developmental and was airborne. Aircraft X was on flight following and assigned runway heading per D10 prior coordination. The aircraft transponder was not observed. I instructed the pilot to ensure transponder on and to squawk as assigned. I observed a splat associate with the aircraft target; slewed to the splat and ensured that it was displaying the correct code. I instructed the pilot to contact Regional Departure. As I reached for the land line to coordinate the untagged aircraft with the D10 Controller I observed a tag display with Aircraft X. Since the aircraft did tag within 2 miles of departure end I had no reason to coordinate and did not do so. I had no other immediate duties involving the RADAR and turned my attention away from it. As I turned back to the RADAR; I heard DAL Tower initiate coordination with me to find out about this aircraft and state that D10 would not answer the line. I observed what appeared to be Aircraft X turned back toward ADS and not displaying a tag. I observed the squawk code momentarily associate with the aircraft target. I observed the data tag for Aircraft X associate with the aircraft target. I observed the aircraft RADAR target and had visual acquisition of the aircraft entering ADS Class D Southeast of ADS airport. The aircraft then turned West (South of ADS airport) and exited ADS Class D. During this aircraft re-entry to ADS Class D I did not attempt any coordination with D10 because (1) DAL Controller had already told me that D10 would not answer them. (2) I could see the aircraft and it did not appear to be in any distress situation (3) I had no imminent or potential situations the aircraft could affect. (4) The aircraft was once again turning away from ADS. (5) I did not want to add any more complexity to what already appeared to be a difficult situation for the D10 Controller. I have noted on previous occasions that a departure off ADS will acquire the data tag for a few seconds and then disappear. At some point the data tag will reappear. I am assuming this is what happened in this instance. This on going problem seems to be random. It needs to be determined what the problem is with tags acquiring; then dropping; then re-acquiring and fix the problem.