AstroTurf was not invented for the Astrodome. It already existed, under a much duller name.
Monsanto researchers developed a synthetic playing surface and called it ChemGrass. Today I Found Out credits Donald L. Elbert, James M. Faria and Robert T. Wright, and dates the invention to 1964. Wikipedia names Faria and Wright as inventors, credits Elbert with later improvements and dates the patent to 1965. The first, experimental installation came in 1964 at the Moses Brown School in Providence, Rhode Island.
Then Houston had a grass problem. Its new domed stadium, the Astrodome, first tried natural grass. The roof panels did not let in enough sunlight, and the grass died within a few months. The team painted the dirt green for a while.
By the start of the 1966 season the Astros had chosen ChemGrass. Supply was short, so only the infield was covered for opening day. The outfield followed in early summer, and the stadium drew national attention.
That publicity gave the product a new name. John A. Wortmann, a Monsanto employee, rebranded it AstroTurf. Etymonline records the trade name from 1966 and explains it as astro- plus turf, after the Astrodome.
Both halves are old. Astro- comes from Greek astron, "star," and English has used it to build words since the mid-1700s. Etymonline notes that Houston was also home to the space program's mission control. Turf is Old English, meaning a slab of earth and grass.
The brand then became a metaphor. In 1985 Texas senator Lloyd Bentsen received a flood of letters and postcards sent on behalf of insurance interests. He dismissed it in the Washington Post: a fellow from Texas "can tell the difference between grass roots and Astro Turf." Word researcher Barry Popik finds this the earliest use of the political sense, and Bentsen is usually credited with it. "Astroturfing," for a fake grassroots campaign, appears in print by 1993.
The product itself has changed hands many times. In 1987 Monsanto consolidated the business as AstroTurf Industries, based in Dalton, Georgia. Since 2016 it has belonged to SportGroup.