Really, as far as specific intentions go, it depends on the intention. For example, praying for it not to rain may be purely selfish(because I don’t like the rain) or another matter entirely(because rain again would cause immense flooding and ruin the homes and lives of many people). Obviously, asking God for a Mercedes Benz isn’t typically a good reason for prayer, but I could imagine instances where asking God for a car would be beneficial to more than just the person doing the asking.
We just have to remember that Christ is not only our Lord and King, He’s also our greatest Friend. I talk about all kinds of things with my friends- what I’ve been struggling with, my thoughts about other people, my future, the weather, anything. Just because Christ can CONTROL those things doesn’t mean we shouldn’t mention them.
In all, we CAN ask for whatever we want from Him, and use a properly formed conscience to determine whether it’s something we SHOULD be asking for or not. And always remember the decision isn’t up to us- even Jesus asked the Father not to die: “Father, if it be thy will, take this cup away from me. But not my will, but thine be done.”
It’s the last part of the phrase that we need to keep in mind- that God DOES know best, and one of the greatest things we can pray for is to know His will, so we can conform ourselves to it.
Just some food for thought.