The pattern of Chidori (zigzag) has a meaning such as stronger luck in a game. That is a pun for "taking a thousand fortunes" which has similar pronunciation as that of Chidori in Japanese.
千鳥という柄は千鳥→千取る→千の福をとるなどと語呂合わせで、
勝負運が強くなるなどの意味があるそうです。
The houndstooth pattern also remind Japanese people flying birds. Though foreigners seems to think it looks like a hound dog's teeth.
千鳥格子という柄も、日本人は鳥が飛ぶようだと思ったけども、
外国人は猟犬の牙だと思ったんだそうです。
It is interesting!
面白いですね。