At the opening of The Flies, Orestes's biggest complaint is that he's "too light." He has no memories, he has no commitments, and no real sense of self. This he interprets as a lack of weight. Let's take a look:
Palaces – that's so. Palaces, statues, pillars – stones, stones, stones! Why, with all these stones in my head, am I not heavier? (1.1.95)
You've left me free as the strands torn by the wind from spiders' webs that one sees floating ten feet above the ground. I'm light as gossamer and walk on air. (1.1.97)
The solution, clearly, is to take some sort of heaviness upon himself. While a lack of commitment (which we know from Orestes's "Character Analysis" translates to a lack of a "fundamental project") is presented as lightness, we soon see that crime is interpreted as a sort of heaviness. It starts with Clytemnestra's advice to Electra:
But wait, my girl; one day you, too, will be trailing after you an inexpiable crime. At every step you will think that you are leaving it behind, but it will remain as heavy as before. (1.1.206)
Orestes hears this and takes it to heart. Before long, he's determined that a crime is the answer to his lightness problem:
I must go down – do you understand? – I must go down into the depths, among you. For you are living, all of you, at the bottom of a pit.
I'm still too – too light. I must take a burden on my shoulders, a load of guilt so heavy as to drag me down, right down into the abyss of Argos. (2.1.165-7)
Believe me, [my crime] weighs on my heart like lead. We were too light, Electra; now our feet sink into the soil, like chariot-wheels in turf. So come with me; we will tread heavily on our way, bowed beneath our precious load. (3.1.153)
But our discussion isn't as simple as "weight = crime." By the third act, the idea of weight has taken on a more nuanced and complex meaning. When Orestes speaks to his sister about together bearing their "precious load," he's referring not only to the murder just committed, but also to the burden that is radical personal freedom. Not that these are entirely different – remember it was the realization of personal freedom that allowed Orestes to commit his crime in the first place.