A common practice in ancient Israel was to incorporate god's name when naming a son.
Two of Saul's sons have interesting names: Merib-baal and Eshbaal (A later redactor changed the names to Mephiboshet and Ishboshet to hide the fact that their names contain the word baal.)
That being said, what motivated Saul, a worshiper of Yahweh, to incorporate Baal into his sons' names? Why did Saul's sons' name do not contain "yah", instead of "baal"?
First, just to be clear Meribbaal/Mephiboshet is actually the grandson of Saul (and son of Jonathan). With that out of the way, the honest truth is that no one can know for sure why Eshbaal and Meribbaal were named as such. The Bible doesn't tell us why, and we have no other source.
We can same something about what the word baal means though. Baal is a common word in Ancient Near East cultures. As a common noun, it could mean "master", "husband", or "owner" in Hebrew. It could be used to form complex nouns. In Genesis 39:19 Joseph is described as a "baal of dreams" = dreamer; in Genesis 49:23 we find "baal of arrows"=archers; and in 1 Kings 1:8 we find "baal of hair"=hairy man; among other such uses within the OT.
Naturally, a word that means "master" is a natural candidate to be adapted to describe a deity (just as Lord, a synonym for master, is used to describe God in English). And this is what several ANE cultures did. There wasn't a single distinct god named Baal, but rather several different gods with that name or derivatives of it. Here is what Encyclopedia Britannica, hardly a religious apologetic, has to say about the use of the term in Israel:
In the formative stages of Israel’s history, the presence of Baal names did not necessarily mean apostasy or even syncretism. The judge Gideon was also named Jerubbaal (Judges 6:32), and King Saul had a son named Ishbaal (I Chronicles 8:33). For those early Hebrews, “Baal” designated the Lord of Israel, just as “Baal” farther north designated the Lord of Lebanon or of Ugarit.
In other words, in Saul's time (11 century BC), it was simply a term used to talk about God just as we might variously use "Lord", "Father", "Yahweh", etc. and yet not be referring to multiple different gods.
This changed, again according to Encyclopedia Britannica when Jezebel attempted
in the 9th century BC, to introduce into Israel her Phoenician cult of Baal in opposition to the official worship of Yahweh (I Kings 18). By the time of the prophet Hosea (mid-8th century BC) the antagonism to Baalism was so strong that the use of the term Baal was often replaced by the contemptuous boshet (“shame”); in compound proper names, for example, Ishbosheth replaced the earlier Ishbaal.
So no, there is no real evidence to suggest people in Saul's time with "baal" as part of their name derived it from Baal worship. Those names could mean "master of X", or be a reference to Israel's master (Yahweh), without needing to invent a contradiction in what the Bible teaches about the history of Israel.
The answer to this can be found in the non-biblical record. Edward F. Campbell Jr. writes in 'A Land Divided: Judah and Israel from the Death of Solomon to the Fall of Samaria', published in The Oxford History of the Biblical World, page 212, about excavations at Samaria that produced over a hundred ostraca (inked notes on sherds) dating probably from the first part of Jeroboam II's reign (roughly 770 BCE). He says that from these records scholars have reconstructed a roster of the names of Samaria residents. A large minority of them are compounded from the divine name Baal, although the majority are Yahweh-compound names.
Another example is Gideon, whose real name was Jerubaal ('May Baal contend'), but which was changed in some references for theological reasons to Gideon, which is based on the root of גָּדַע (gada, 'to hew') and thus means 'The one who cut down', a biblical reference to cutting down the asherah.
According to the Bible, Saul had a son called Eshbaal ('man of Baal'). Meribbaal ('Baal is my advocate') was the son of Jonathan (1 Chronicles 8:34;9:40). It is likely that Yahweh was one of the gods that Saul worshipped, but more importantly, Baal was one of the Hebrew gods during the time of the early monarchy. Mark S. Smith says, in The Early History of God, page 64, that on the available evidence, Israelite religion in its earliest form was polytheistic and the number of deities was relatively typical for the region.
Please note that God did not desire the title baal, as previously stated. That would go against the nature, holiness, and position of Yahweh. In Hosea 3:17 (NKJV) God says: "For I will take from her mouth the names of the Baals, And they shall be remembered by their name no more." This showed God's displeasure with Israel for being unfaithful.
