Sva: One's own, belonging to oneself, ego
Ha: HÄ (हा). To drive away, lose, dispel
The Puranas tell the story of Swaha.
In short, the Gods were hungry, and informed Lord Brahma that they had nothing to eat.
When Brahma pointed out that their food was the oblations poured into fire by humans during yagnas on earth, Agni (The god of fire) confessed that he did not possess the agency to make that transfer possible. So no offerings would make the journey from Havan to the Gods.
Brahma consulted Prakriti, who created a Goddess who would be the cosmic conduit to ensure that the offerings made on earth reached the gods in heaven.
According to the Brahmavidya Upanishad, Svaha represents the Shakti, or power, that cannot be burned by Agni.
In the Upanishads, Svaha confesses to be enamoured by Agni and wishes to dwell with him. Since the name of the Goddess was Swaha, Brahma decreed to Agni that invoking her name during hymns, would allow Svaha to dwell with Agni in perpetuity, and only then, would oblations poured into the fire reach the gods.