If the following statements are to scientific, just forget about them. They are about the etymology of Saul, which is indeed a bit tricky.
The board states “asked for” or “prayed for” as the meaning of Saul. In Hebrew it’s “Sha’ul”, and the verbal root is “sha’al” (Shin-Aleph-Lamed; always three consonants in Hebrew verbs unless one is dropped). Now the first meaning of the verb is indeed: “to ask (for)” or “to pray for”, so the translation given above is correct, and that’s what you are likely to find in most name books. There is also a sufficient name giving motive: “I prayed to God to receive this child.” But there is also a different meaning to the name, favoured by many Old Testament scholars, and this has to do with another Saul mentioned in the bible: Saul, son of Kis, the first king of Israel. The bible does not give an etymological explanation of his name (nor of Shaul, an Edomite king mentioned in Gen 36,37f., nor of two other persons by that name, Gen 46,10 and 1. Chr 6,9), like it does in many other cases. But strange enough, you will find this explanation at a different place, namely with the birth of Samuel, the prophet and judge. In 1. Sam 1,20 his mother Channa says: “I will call him Samuel, because I prayed for him.” But this is the explanation of the name Shaul, not Samuel. Now what does the name Samuel actually mean? – We don’t know. Often you will hear “God has heard (and answered)”, and this would at least match the meaning of Shaul given above: I have asked – God has answered. But the translation is doubtful, there is one consonant (see above) missing, the Ayin from “shama” (Shin-Mem-Ayin) meaning “hear, listen” (it could still be a contraction, though). Mostly it is translated “name of God” or “(E)shem is God”, (E)shem being a pagan god, we don’t know much about. Now Channa says even more things obviously related to Sha’ul, not to Samuel (v. 27): “I prayed for this boy, and the Lord has given me MY ASKING, which I ASKED FOR from him.” (more or less litteral translation, in Hebrew it reads: SHE’ELTI asher SHA’ALTI, just to give you the similarity to the name SHA’UL) And she goes on: “And also I will LEND him (hiSH’ILTIhu) to the Lord all the days, that he will be LENT (SHA’UL) to the Lord.” Here the verb “sha’al” obviously has a different meaning (to lend or to return), and there seems to be some agreement, that this should be the original meaning of the name Sha’ul. But I don’t think many mothers had this in mind when they named their boy Shaul. So you may well stick to "asked for, prayed for". Why this explantion is given at the wrong place in the bible, nobody seems to know; at least I couldn’t find anything about this.
(I am basically following: Martin Noth, Die israelitischen Personennamen im Rahmen der gemeinsemitischen Namensgebung, Stuttgart 1928, pp. 123, 136 note 2; despite its age still the standard work on the subject)