Who designed and built the T-800 series Terminator endoskeleton prop?

Joseph P. Lucky was the art director for T2. George Costello was the art director for T1. However, the man most directly involved with the construction of the Terminator props was Stan Winston, and he is credited as Terminator 2's special makeup producer and T1's special terminator effects creator, at IMDB.

“The first Terminator robot was made of a plastic material,” said 25-year SWS supervisor & Co-Founder of Legacy Effects, Shane Mahan, “like a lens cap that might have the look of chrome, but is really plastic. We’d run the robot pieces through an electrostatic process to apply a metallic finish; but, in shooting the first Terminator, we’d found that it chipped very easily. That was a heavy action film — as this one would be — and we were constantly bashing that thing through walls.

So, by the end of shooting Terminator, the endoskeleton puppets were literally patched together with paint and tin foil. There were little patches all over them to hide where the metallic finish had flaked off. By the time we got to Terminator 2, we used an actual chroming process for making the endoskeleton. It was a heavier material, but it made the endoskeleton puppets more durable, and the metallic luster was much more authentic looking. It made a huge difference.” –stanwinstonschool.com

(I invite you to check out that link above. It's a great read and has lots of cool pictures ;)

James Cameron made the illustration from which Stan Winston took his inspiration. So, they were "designed" by Cameron and built by Winston and his studio.

Original concept art by James Cameron :

Below on the left, an early painting by James Cameron, next to the image from the movie. On that painting, the endoskeleton design is virtually identical to the final version. This is one of the paintings that Cameron brought to Stan Winston. "Jim is an artist. He came to me and had the script, but more importantly he had the drawings of what he wanted things to look like." (Stan Winston Starlog 88, 1984).

"I remember Stan and Jim going off to an auto part graveyards and took photographs of interesting pieces of machinery for reference." (Shane Mahan, Winston Effect)

Still, Cameron was open to suggestions and wanted to see some choices, so he encouraged Winston and his team to come up with their own ideas. "Jim wanted to see some other variations. He's a brilliant artist but he wanted choices. So we went through some variations, some sculptures of different possibilities of the look of the endoskeleton and we ended up going right back to the initial concept of Jim he had in the painting he did." (Stan Winston) –JamesCameronOnline.com