Hmm, well let's take a look:
Hoenn Regirock:
* Location: Desert Ruins
* RS & ORAS Solution: Right, Right, Down, Down, Strength.
* Emerald Solution: Left, Left, Down, Down, Rock Smash.
* Interpretation:
My assumption for the chamber locations isn't that they're representative of the Regi they contain, but rather they were picked specifically to KEEP the Regi sealed there. Yes, a desert with a constant blowing sandstorm seems like a fitting place for Regirock, but Regirock's weight would likely cause it to sink into the sand and be buried or risked prolonged exposure to the eroding sandstorm.
Now onto the puzzle. For the steps portion, since you're walking on a rock floor, maybe it's not the exact step directions that matters but following an angular "rigid" pattern; like a rock. May also in a way bring to mind a corner of Regirock's eye pattern which are dots in an H shape. Using Strength & Rock Smash is more easier to understand: Regirock is the physically strongest member of the group. Those who want to release it would have to show they're just as strong as it, likely the spot where you perform the moves on being a pressure plate that has to be pushed down or broken.
Hoenn Regice:
* Location: Island Cave
* RS & ORAS Solution: Wait 2 minutes.
* Emerald Solution: Walk around the wall for one lap.
* Interpretation:
Being a tropical region, there's not a lot of cold locations to store Regice. Best place would be underground, with the added bonus the tropical heat would keep Regice from wanting to escape. As for why Route 105, well if it did escape it would have to float in the water to the mainland, and Route 105 is noted for "gentle" currents so it would be a very slow float under a hot sun.
For the RS & ORAS puzzle, of the 3 original Regi, Regice would likely be the most affected by time. While essentially magic keeps it from melting as the dex describes, symbolically rock and metal can stay pretty sturdy for centuries even under non-favorable conditions; but unless ice is in the absolute perfect condition it'll start melting as soon as it the temp goes above freezing. Never timed it myself, but 2 minutes for an ice cube to melt in room temperature sounds about right, so I guess you're symbolically waiting for the "ice to melt" to face Regice.
In Emerald, wanting to work around a potential second "frozen clock" glitch, changed the puzzle so it wasn't time based. Too meta? Okay, so how about this: when something freezes it's not immediately encased in ice but rather layers of frost travels across the surface. So by walking around the edge of the chamber you're acting like a layer of frost which leads to a block of ice.
Hoenn Registeel:
* Location: Ancient Tomb
* RS & ORAS Solution: "With love, hope, and time" Stand in the center and use Fly.
* Emerald Solution: "To inherent their will" Stand in the center and use Flash.
* Interpretation:
The water surrounding the chamber hints this is a place that frequently gets a lot of rain. Best way to keep a metal titan inside is if the outside woud likely make it rust and/or make the terrain muddy which it'll sink into and get stuck.
Now Registeel's puzzle is more interesting as it's more poetic in its presentation where the other two were straight forward instructions. This is probably because Registeel is the most "artificial" of the original Regi, it looks manmade (whereas Regirock cobbles itself together with loose rock and Regice is a giant ice block). I see this puzzle being a bit more "spiritual" to relate to that artificial/manmade theme. Love, hope, knowledge & observation of the passage of time (as well as mortality), and willpower are all "concepts", things which humankind has made and gave meaning to.
Why is knowing this important? It explains the move you then got to use. In RSE it's Fly, while concepts don't physically exist, artistic interpretation often present concepts as floating spirit-things. To be filled with love, hope, and time is to be in peace with oneself, giving a feeling of freedom, using them to lift oneself up and "soar into freedom". In Emerald, they just use "will", or rather "inheriting will". Who's will? Guessing the ancient people, the "will" you're inheriting is likely their knowledge and possibly their trust. We all are inheritors in many way: we inherit our genes from our parents, the world & its knowledge from the previous generations, and on a smaller scale many inherit conceptual & material things. And we use these inherited "wills" to make ourselves better, you could say to "make us shine".
And once you've mastered these human feelings, it's time to face something which isn't very much human in the slightest yet still feels like it could have been a result of it.
Sinnoh Regigigas:
* Location: Snowpoint Temple
* Solution: Bring it the 3 Regi.
* Interpretation:
Only thing to interpret here is why did Regigigas choose to rest in Snowpoint. Looking at the Sinnoh Map, Snowpoint is the most North place on the Sinnoh mainland. Only way to get up to Snowpoint it through Mt. Coronet, and I just don't mean walking through a simple cave passage but rather you do have to do a bit of spelunking to get onto the level where the entrance to Route 216 is. I'm not sure exactly what this all means, but I don't feel it's a coincidence. Could be after moving around the continents and notably Mt. Coronet, Regigigas went someplace where it could observe its work and, satisfied, entered hibernation until it was time for it to awaken again (or be awakened). Infact that could be another reason why its so far away from the center of Sinnoh, if anything major happened to Sinnoh it would likely be on or around Mt. Coronet and Regigigas being in Snowpoint puts it in a position where it's both safe and can easily get to it.
Unovas Underground Ruins:
* Regirock Solution: To access the Rock Peak Chamber you'll need to get the two step directions from the "eye" in the top center of the room: during the day you're told to go 6 down, at night you're told to go 9 right.
* Interpretation:
Regirock is the only one you can access in both games. It's puzzle is similar to its original needing to walk a certain number of steps in certain directions (at least this time you don't need an additional move to click on the spot to open the chamber). As before, one symbolism if you're navigating an angular path on a rocky terrain. The numbers, 6 and 9, and more open to interpretation aside random numbers so the player doesn't easily guess where the tile they need is (or maybe that's all there is as I can't think of anything).
ORAS Regigigas:
* Location: Island Cave
* Solution: During the day, also having Regirock & Registeel in the party, have a nicknamed Regice holding a "cold item" and go to the center of Island Cave.
* Interpretation:
... So let's break it down.
Having the 3 Regi makes sense for a requirement to get Regigigas.
Why Regice and it's chamber? Well, maybe as a reference to Snowpoint and Sinnoh? Regice is Ice-type which is a fitting representation for Snowpoint and Sinnoh is an island like the Island Cave (though obviously on a much bigger scale).
Regice holding a "cold item", once again maybe a reference to Snowpoint. Snowpoint not only was a cold location (which was shaped like an 8-pointed snowflake (FUN FACT: due to the property of water molecules real snowflakes are 6-pointed)) but it was also the location of the Ice-type Gym Leader. REALLY hammering in the "snow/ice" theme.
Nicknaming Regice.
Because GF just wants to f*** with you.
Because it's agreed Regice's name breaks the naming theme and they want to see you do better.
While it's any cold item in all honesty the Regis just want ice cream; Regice is holding it to keep it from melting. It's interesting that, considering what they're based on, changing their names or giving them something like Mail with a message hasn't been done before. The Regi are based on
the golem of Hebrew mythology. The golem was made of clay and meant to be a guardian figure, the Rabbi who made it controlling it with written symbols that it was fed on paper or engraved on its face. That's why they have the dots on their face and connected to a braille puzzle. But maybe this could also be why you have to give Regice a nickname, there a specific reason why Regigigas placed the patterns as it did. By you even giving it a nickname you risk altering the meaning thus making Regice unable to serve its duty. Island Cave must have a connection with Snowpoint Temple, and upon sensing the Regice in the Island Cave have been altered with it manifests itself into the Island Cave to do battle with whoever is messing with its work. Now it would have just been as angry if we did the same thing with Regirock and Registeel, but they don't have a connection with Island Cave thus Regigigas can't sense any tampering to them.