The royal widow was a powerful figure in both Argead and Persian courts, traditionally neutralized by the successor or regent with marriage (either to her or her daughter). The unstable structure and sheer size of Alexander's empire, the multiplicity of war-hardened grandees and the uncertain succession to a new entity (not that the Argeads and Achaemenids didn't have their own succession dramas) only to serve to complicate the situation further. Allowing these women to remain unmarried or be distributed among Alerxander's generals is way too dangerous - far more dangerous than marrying them himself as a means of concentrating legitimacy in his person and eventually sidelining the Argead males as Alexander's successor.
Marriage with Roxana would secure Hephaestion's position as Regent for Alexander IV; marriage with Stateira consolidate him as the legitimate King of Asia; marriage with Cleopatra legitimize him as King of Macedonia and Epirus and "son" of Olympias. A bloodbath would inevitably follow but I'm confident in Hephaestion and Olympias' combined ability to prevail over the rest.
The question begs itself: if this is such a good idea, why didn't it happen IOTL? Perdiccas was, essentially, in the role we're casting Hephaestion in here. After Alexander died, Roxane grew nervous that Stateira or Parysatis would suddenly become pregnant and thus be able to challenge her baby as Alexander's successor. She then collaborated with Perdiccas to have both killed. Perdiccas never aspired to make Roxane his wife, working to protect her and Alexander IV but never once trying to become the boy-king's step-father, even while he set aside his Susa bride (as I recall, the daughter of Atropates) and chased Cleopatra and Nicaea trying to find the best alliance. As I recall, polygamy was only practiced by kings in Macedonian society. Note that most of Alexander's successors never picked up the practice, and that the most obvious one that did, Ptolemy, only did so once Alexander was dead and his position secure and unchallenged in Egypt. He married Antipater's daughter first, then a few years later married Berenice, all the while maintaining the Athenian prostitute Thais as a de facto but not de jure consort. I could be wrong on this, but I think it was the case prior to Alexander's death - I can't think of anyone besides the king having more than one legal wife at a time in his army or in Philip's army before that point.
If Hephaestion takes Roxane and/or Stateira and Parysatis, he is essentially declaring himself king over Alexander's brother and child, almost to the point of declaring himself to be Alexander. This wouldn't be horribly surprising given his history with Alexander. However, in other Argead/Achaemenid examples, the regent was always an Argead or Achaemenid and had a claim to the throne himself. Hephaestion is a minor noble who owes his entire position to being Alexander's (probable) lover and a competent general in his own right. He wouldn't be allowed to declare himself king so easily, not at this stage. While he might be ruling in fact, his many enemies would surely capitalize on the obvious grasps for power in these moves. Perdiccas' story was one of compromise, trying to keep a huge number of factions pleased, until it turned out no one was happy with him (except Eumenes) and finally he picked a side and lost. It didn't work, but blatantly picking one side won't work much better than Perdiccas' blatant decision to pick Olympias unless of course he actually wins, unlike his OTL counterpart. And I'm not sure that Hephaestion and Olympias together are stronger than Antipater, Craterus, Perdiccas, Ptolemy, etc. etc. Hephaestion was not the most beloved officer in the army (Craterus was), and Antipater holds Macedonia. He wasn't as good a general as either of those men.
Even if Hephaestion marries Roxane, Stateira (somewhat redundantly as a sister of his wife Drypetis, but you do have a point that it does prevent another from trying to marry her, although a man in Hephaestion's position should be able to prevent that anyways), and Parysatis, I feel it's almost certain that Roxane seeks for help in eliminating all of the women who might be able to claim their son has a better claim than her own to the throne. Hephaestion wouldn't do it, being married to all of them, including Darius III's two daughters. So she must seek his enemies. Working with Craterus or someone else, she eliminates at least Stateira and Parysatis like she did IOTL, and maybe Drypetis too, if she's extra paranoid (and I think there are rumors that Drypetis was part of the purge anyways). There goes all of Hephaestion's nice claims. Or, she uses her newfound closeness with Hephaestion to murder him (preemptively protecting herself), and then go on to purge the other wives and protect her position as the next king's mother.
There are only two ways that this goes well for Hephaestion: one, he outschemes Roxane and has her incapacitated before she kills Alexander's other widows and his wife (or himself), and then somehow manages to gain the support of enough of the army and defeats Craterus and Antipater and Antigonus and anyone else that might rebel; or two, literally have Alexander's dying wish to be for Hephaestion to marry his wives and proclaim his unborn son with Roxane to be his heir, which should be enough to calm most, at least for the time being.
In short, I don't see this scenario working out well for Hephaestion or for the preservation of the empire. I'm not wholly sure whether it would have been accepted for Hephaestion to take more than one wife, but taking on all of Alexander's widows would have been seen as the blatant power grab it would have been, and the already hostile aristocracy would have been glad to see a reason to pounce. Hephaestion would have to maintain his crown with force, and there isn't much reason in my view to see how he would've been successful without most of the key members of the aristocracy supporting him, those aristocrats usually being seen as greater generals than him. It's not that complicated to make sure that Alexander's widows don't marry one of his rivals - just keep them secure in his court under close supervision. He's the one in control here, he'd have the power to do that. He doesn't have to marry all of them himself. And even if he wins a civil war, he's have a great deal of reconquering to do in the east, which if it's anything like OTL will have fallen from Macedonian control.
Even shorter: Hephaestion probably isn't the one to keep the empire together. He's too divisive and probably not strong enough to maintain order or to conquer it back. Seleucus had a much better shot forty years later, even in his late seventies, to at least rule the entire empire for a year or two before that likely falls apart simply due to the empire's sheer magnitude.
EDIT: I forgot to make one other point, that I meant to make at the beginning: marriage alliances with easterners weren't worth nearly as much after Alexander died IOTL. Hephaestion as a supporter of Alexander's integration policy might have tried to preserve that policy, but really the old Achaemenid court didn't provide much power once Alexander had purged the eastern satrapies of old Achaemenid loyalists, and the death blow was when nearly all of Alexander's generals ditched their wives in favor of Greek and Macedonian wives almost as soon as he died. He's going to be dealing almost entirely with Macedonians, and its with Macedonians that Hephaestion would need to make allies with in order to preserve the empire, not Persians. Perdiccas never once considered marrying Alexander's wives. Why? In part I think for the reasons I've already stated (no polygamy for non-royals IIRC, assumption of too much symbolic power, etc.), but also because they really don't actually improve his position that much. If he thought that marrying the last Achaemenid princesses would strengthen his claim to be "Lord of Asia", why did he assist Roxane in killing them? And if he was working closely with Roxane, why didn't he marry her instead of worrying about Nicaea or Cleopatra?