Why did this discount store adopt that name?
According to their website because “it is the first positive day of the week.”
When the chain began,it wasn’t open year-round, but only had special sales events four times a year, each opening on a Tuesday morning. The founder picked the name: "Naming the company was easy. Ross had wanted to run a company named after a day of the week ever since he was a teenager delivering copies of the Saturday Evening Post. He picked Tuesday because in his opinion, “Tuesday morning is the first positive day of the week,” according to company press materials. "
Later, when they did set up permanent locations, Tuesday mornings were the time they stocked the new merchandise, so it was the “best” time to get first pick of what was available. That schedule was ditched sometime in the 1980s or perhaps very early 90s; now, they’re open all week long and the day(s) when new merchandise arrives varies by location.