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.