This name can be used as 寧子, 根子, 音子, 子子 or 眠子 with 寧 (nei, mushi.ro) meaning "preferably, rather," 根 (kon, ne, -ne) meaning "head (pimple), radical, root," 音 (in, on, -non, oto, ne) meaning "sound, noise," 子 (shi, su, tsu, ko, -ko, ne) meaning "child, sign of the rat (1st sign of the Chinese zodiac)" and 眠 (min, nemu.i, nemu.ru, ne) meaning "die, sleep(y)."
It coincides with the word (猫) meaning "cat," though it's not used as a name in Japan.
One bearer of this name is lyricist Neko Oikawa (及川 眠子), who wrote the theme song for Ultraman Max, among other things, though it's not likely that's her real name.
This name is very rarely given to girls, if given at all.
Neko (根子) is also used as a surname.