Why does my buttercream become runny overnight?
Meringue is a protein-based foam. The proteins form a semi-stable mesh with air bubbles trapped inside. The reason they are semi-stable is that they unravel a bit and hook into each other, Velcro-style. When you introduce fat, it lubricates and smooths the unraveled proteins, so they lose their hooking ability. If the fat is introduced before the eggwhites are beaten, there is a fair chance that you won't get a foam at all. 

It is possible to mix the prepared foam with fat without destroying it, although it will lose some volume. This is usually done when the bulk of the mix is not the meringue, but some kind of creamy filling which gets lightened by the air added through the meringue. These fillings are usually set through baking (such as a cheesecake filling), or gelling agents (there are cremes which combine egg whites foam, fat-containing dairy products, and gelatine). As you did not set the icing in any way, the fat from the butter slowly worked itself into the formed hooks of the proteins, destroying the mesh and returning the eggwhites to their original form (a suspension of proteins in water). Eventually, if you overbeat the meringue a bit, you also got some weeping caused by the proteins slowly getting tight enough to expell some water from the structure, but if present at all, this is probably a secondary effect in this case. 

Bottom line: the mix you made is inherently unstable. I wouldn't use it as an icing unless the food is eaten quickly. You should get a different recipe. If pure buttercream is too rich for your taste, you can try versions which add yolks or whole eggs (not foamed eggwhites). The lecithine in the yolks emulsifies the rest of the egg with the butter, so you don't get separation problems. Another option is to forego buttercream completely and use whipped cream, it still gives you a rich mouthfeel, but has much less fat per weight and whips to a higher volume.