I wouldn't say that it "minimizes" the request; I'd say that it intensifies the focus of the request.

Yes, we can use just to emphasize how small something is. For example, a salesman might say:

I'm just asking for a minute of your time.

but I don't think that's the same as when we are talking about God's mercy in a prayer.

Sometimes, we're not trying to say that something is small. Instead, it's more akin to when someone says:

I just want to thank all those who helped in the rescue efforts

The word "just" serves as an intensifier there. We could even take it out of the sentence without changing its meaning:

I want to thank all those who helped in the rescue efforts

Moreover, it certain isn't meant to diminish the accomplishments of the rescuers – not in the same way the salesman is trying to trivialize the amount of time he was requesting.

As @stangdon says in a comment, it's more about focusing on the moment.

Wordnik offers an interesting meaning, among several others for just:

just (adv.) Used to show humility

I think this is why we see this word used often in prayers and commencement speeches; people are humbled by all they have been given, and they just want to express their heartfelt gratitude.

The general form of your question is

What is the meaning of 'just' as an adverb?

The OED lists 4 main senses in current use, 2 obsolete senses, 11 phrases, 2 compounds. All of them derive from the adj. 'just' from Latin 'iustus' ("lawful") but, since English had the separate word 'justly' that feels more appropriate for all the "lawfully" and "appropriately" senses, the adv. 'just' came to focus on the sense meaning "exactly" or, concerning verbs in the past tense, "almost exactly before".

From there, you get

5. Limiting the extent or degree denoted by an expression: only as much as, not much more or less than; barely, by a little, by a slight margin.
...
6.... a. No less than; absolutely; actually, positively, really. In weakened sense: neither more nor less than, no other than; simply, merely.

In other words, it became a synonym of "merely" or "only [the thing I'm about to describe]".

Therefore, in your prayers

Lord, we just want to ask for your grace...
G-d, we just want to thank you for the gift of Salvation...

the effect of the 'just' is to underline that you want the Almighty to believe—rightly or wrongly—that no ulterior motive prompts your action. The only thing you wish to do is request grace, not defend yourself or pretend to blamelessness or offer mitigating excuses for your many sins against His Law. The only thing you wish to do is thank Him, not request a favor or ingratiate yourself or put on a pious show for the community.

It is a rather self-defeating turn of phrase. It instantly brings to mind Danish noblewomen "protesting too much" or liars throwing in a "you gotta believe me" or "when have I ever lied to you" that instantly reminds you of several instances. Someone who is honestly just asking would, in fact, just ask. One might easily contrast it with the direct—even blunt—phrasing of, e.g., the Lord's Prayer offered by the guy who was so pissed at the Pharisees' showy piety.

On the other hand, as a Christian prayer, it could easily be taken as instructive or aspirational. The dogma goes that—with very few and explicitly listed exceptions—mankind is born, lives, and dies in sin and their forgiving Father knows the deceit and self-interest that prompt most prayers. A truly pious person wouldn't pray this way, but a self-aware sinner might. Similarly, a church might enjoin it, intending that the laity should be reminded that Christians should try to overcome the many limitations to their kindness and love, beginning with their relationship to the Almighty.