🔼The name Shimrith in the Bible
The one and only Shimrith of the Bible is the Moabite mother of Jehozabad, who conspired with Zabad (or Jozabad), the son of Shimeath the Ammonitess, to assassinate king Joash of Judah (2 Chronicles 24:26). In the parallel text of 2 Kings 12:21 Jehozabad is said to be a son of Shomer, and it's most plausible that Shomer was not his father but his mother. The names Shomer and Shimrith are closely related.
🔼Etymology of the name Shimrith
The name Shimrith comes from the root group שמר (shmr I, II & III):
How this name ended up with a yod-taw ending isn't clear, but this name was constructed long before this story was written. And back then, the letter taw sometimes existed where in the younger Hebrew of the story, the letter he occurs. This name may thus come from the noun shomra, which was then shomrath, and the yod may have been inserted for flavor. With a similar theory the name Elath may be explained.
For a meaning of the name Shimrith, NOBSE Study Bible Name List reads Vigilant. Jones' Dictionary of Old Testament Proper Names proposes Guarded. BDB Theological Dictionary doesn't translate this name but lists it under שמר I.