🔼The name Shaalim in the Bible
The name Shaalim (say: Sha-alim) occurs only once in the Bible. When the donkeys of Kish of Benjamin go missing, his son, the future king Saul, goes looking for them, and passes the land of Shaalim (1 Samuel 9:4). It's not clear where precisely this "land of Shaalim" might have been, but it may have been the same as the "land of Shual" mentioned in 1 Samuel 13:17, and both may have to do with a place called Shaalabbin or Shaalbim in the territory of Dan (Joshua 9:42, Judges 1:35).
Dan's land was originally to the north-east of Judah and to the south of Ephraim and Benjamin, but after Dan's migrations (Judges 18:1) and the decimation of the tribe of Benjamin after the atrocities committed in Gibeah (Judges 19-21), the borders had shifted somewhat.
After his futile donkey hunt, Saul ends up in an unnamed town with Samuel, who normally held seat in Shiloh in Ephraim, and it's quite plausible that the land of Shaalim was named after the Danite town of Shaalabbin it contained.
🔼Etymology of the name Shaalim
The name Shaalim is the same as the plural form of the Hebrew word שועל (shu'al), meaning fox or jackal:
For a meaning of the name Shaalim, NOBSE Study Bible Name List reads District Of Foxes. Jones' Dictionary of Old Testament Proper Names and BDB Theological Dictionary equate Shaalim with Shaalabbin and Shaalbim, for which they read Place Of Foxes (Jones) and Haunt Of Foxes (BDB).
Note the obvious pun of 1 Samuel 9:4, where dashingly handsome and tall king-to-be Saul goes clumping after a herd of donkeys. How hard can it be to find a herd of domesticated donkeys that leave tracks wherever they saunter? Obviously very hard for Saul, who wasn't only very handsome, he was also not very clever. He and his entourage first cross the land of Shalishah (perhaps saying that Saul was able to confidently count to three) and then traverse the land of Shaalim (the intellectual capacity of a fox).
When they finally arrive at Zuph (Honeycomb; pay dirt) Saul asks and asks until at long last he comes face to face with the patiently waiting Samuel who promptly anoints him king over Israel (1 Samuel 9:16).
It seems that the author is saying that Israel stumbled onto monarchy like Saul stumbled onto Samuel, mostly by accident (that is to say: divine providence) and certainly not by any measure of intellectual prowess. The donkeys Saul was searching for had been long found and brought back home by folks who had stumbled upon them (9:20).