M98 Controller described the development of a unsafe situation when a departure would not tag up due to duplicate flight numbers; diverting needed attention from other operational requirements.

Narrative

Air Carrier Y departed during a departure rush; I was unable to track start the aircraft. I tried several different ARTS entries; but had no success. I finally discovered that an Air Carrier Z was also inbound to MSP with the identical call sign approximately 20 Northwest of Minneapolis. I then called ZMP to coordinate the manual hand off; and coordinate the inactive flight plan on the departure. During this time frame; departures continued to launch and I was dealing with weather to the southwest of the airport. I had climbed an A320 to 170 and issued a 225 heading to go outside another aircraft I had climbing to 120; this put Air Carrier X closer to the weather then he liked; normally I would have had time to explain to him my plan and how it would work reference the weather; but I was distracted working the Air Carrier Y issue. Air Carrier X called and said he needed a left turn for weather; I had to unable his request due to traffic. This dispatch problem of putting two aircraft in the NAS at the same time with the same call sign appears to happen at the worst of times. The airlines should have safeguards in place that automatically prevent this sort of thing. I have reported the same problem; here we are one year later; and the problem remains. Recommendation; coordinate with airline dispatchers to ensure safeguards are in place to prevent this from happening.