This is not really a theory. Just some thoughts.
I'll start with something that some may consider off-topic, or probably it is something painfully obvious and well-known, but I can't help but include it here. According to this (or probably also some text-based sources but I honestly don't recall seeing this bit anywhere else) Atium alloys have mental and temporal effects while lerasium alloys have enhancement and physical effect. Now, my point is: I have been a little bothered when the perfect binary pattern (physical-mental, inner-outer, pull-push +2-3 "noble" metals that were supposedly different) was kind of broken when the metals were split into four arbitrary groups. However, if there is some connection between the mental and temporal metals and enhancement and physical metals, we get back to binary tree again by splitting them into "atium family" and "lerasium family" and then into "common metals" and "rare metals" to get the same 4 groups that are suddenly not arbitrary at all!
Now to the specific god metals:
I see frequent theories about its connection to electrum, even some stating it is actually its alloy or otherwise related. For a while I also thought it, but now I don't. Yes, the gold-malatium atium-electrum pattern is sure interesting, but I don't think two metals is enough to set a pattern. The pattern has 2 issues:
How would this pattern extend to other metals? In order not to go too far, let's take another temporal metal - cadmium. What would its reverse effect be? Slowing time instead of speeding it up? That's bendalloy. Speed it up for everyone else instead? That's again bendalloy, just from a different frame of reference. (On a side note, would bronze let me see what metals I am burning? That's almost as beautifully useless as aluminum!) Referring to the same picture again, Atium seems to have an additional effect to seeing the future, that is, enhancing mind. This fits Atium's connection to mental and temporal metals I talked about earlier and also sets it apart from electrum.
To sum up: Atium is a temporal AND mental metal. Its similarity to electrum seems to be just a jolly coincidence. I don't dare guess what the actual atium-electrum alloy would be like, but probably something other than a direct opposite as is case with gold and malatium.
Also note it is related to the Ruin and is the hemallurgic wild card (hemallurgy is of Ruin, according to Sazed in HoA). I am pointing this out because I will refer to it later.
By giving people incredible power, lerasium clearly identifies itself as an enhancement metal. The connection to physical metals is, unfortunately, not as clear as atium's to mental ones. I don't really want to theorize on this because I don't have anything I could base the theories on. Maybe it has some physical effects on body we don't know about.
By turning anyone into a full mistborn, lerasium (related to Preservation) is an allomatic (which is of Preservation) wild card.
Here my frustration partially returns: THREE metals! Not a power of two! And what groups will it belong to? All four are taken! Oh well... I'll return to this later.
For now: feruchemy is, according to Sazed "the power of balance". Harmony is not quite balance, but they are synonymous enough for me to say it will be the feruchemy wild card. What exactly would it do? Universal storage would be in line with hemallurgy and atium, but it would be boring. Maybe it could store feruchemy itself, so perhaps it could be used to bestow one's feruchemy to someone else (it would need to be an exception to the general rule that one can only use their own metalminds, but then, everyone can burn lerasium also). Nothing lost, nothing gained, just transferred.
Now for the patterns: this is very long shot, and I myself don't believe it at all, it's just a silly little thought: there would be a fourth god metal, and harmonium would govern the "rare metals" and the other one the "common metals", or the other way around. Thus, each of the familiar four groups would be an intersection of two god metals. Alternatively, they could be split into "those that govern other metals" and "those that don't". Another alternative: hamonium would be the "ultimate all-ruler" and the fourth one would govern none, giving a weaker pattern, but still a pattern.
I am confused about what wild card would this metal be. It would need to be one, though. My thoughts were: there are three things hemallurgy can steal: allomancy, feruchemy and basic attributes. So this metal would be related to basic attributes? I have honestly no idea how it could work, a powerful vitamin for everyone to eat? Better abandon this train of thought, I think.