As GIT has explained, we can understand what the Bible means when it says He has an "arm" or something like that. The question still stands... What is the Bible trying to say the 20+ times it says God repents? Does God repent or not? Secondly, how do you interpret the passages that say God "does not repent" literally, but choose to interpret the passages that say He "does repent" figuratively?
Here is a passage like the one you mention:
1SA 15:11 I repent that I have made Saul king...
1SA 15:29 And also the Glory of Israel will not lie or repent; for He is not a man that He should repent.
God "repented" in verse 11, in verse 29 we are told he does not repent. In the same situation, God is described as both repenting and we are also told he does not repent. But we have more, here, we are told God does not lie, so we know he is not just pretending to repent (or not repent). Whatever it means, means something, it is not just a show.
Now, we would have an issue if God said, "I have an arm," if He does not actually have an arm. However, God never says that about Himself. Rather, OT authors describe physical attributes of God which we are able to explain the meaning behind. Now, I ask, why would God say He repents, if He does not repent?
It's true there's no Scripture verse that says "God doesn't have an arm". The closest I can think of offhand is Jer. 23:24. But Jesus says in John 15, "I am the true vine." Now we don't need Jesus to add somewhere "I am not *really* a vine."
So your question can be repeated here: "Why would Jesus say he is the vine, if he is not a vine?" I would answer: To tell us something about himself, by way of analogy. I don't think everything about a vine tells us something about Jesus, so we have to understand what parts of the analogy apply, and what parts don't.
In "repenting," in saying "the Lamb of God," in picturing the Holy Spirit as a dove, in all that God describes himself being or doing, we have to recognize that all analogies, all descriptions, are partial pictures (though not false pictures).
I think the passages that say "God does not repent" are intended to tell us that not all of what we do when we repent applies to God. And I think Scripture gives us an idea as to what parts don't apply:
MAL 3:6 I the Lord do not change.
JAS 1:17 ... the Father of lights, with whom there is no variation, or shifting shadow.
PS 33:11 But the plans of the Lord stand firm forever, the purposes of his heart through all generations.
God does not change, nor does he change his plan.
Another point: God oftentimes speaks with us from our point of view:
JN 6:5-6 When Jesus looked up and saw a great crowd coming toward him, he said to Philip, "Where shall we buy bread for these people to eat?" He asked this only to test him, for he already had in mind what he was going to do.
GE 32:31 The sun rose above him as he passed Peniel...
Now God was well aware that the sun didn't actually rise. But he doesn't say "the earth rotated." Similarly with God's actions:
JER 26:3 Perhaps they will listen...
This was speaking from Jeremiah's perspective (see Jer. 36:6).
Similarly with "repents", we can say that God has a change of his response, a change even of emotional response, and speaks also from our perspective here.
And we can understand what he means by our own experience of repenting, it means something like what we experience, but it is also different. Like using our arm...
Blessings,
Lee