Shinigami-sama ni Saigo no Onegai wo

@abos, although your deduction is good, it's wrong. The main point being that even if Shizuka weren't concious/willing to murder, IF Shizuka still commited the murder, he wouldn't be allowed to have a "last request". This was proven when Shizuka battled with the Chief Shinigami (Proven he didn't kill anyone) and as seen in the chapters were the twin evil spirit posess a girl, it's repeated that the girl must NOT be allowed to do the murder, implying she'd have to pay the consequences for the murder.

Now, although I don't have a concrete theory just yet, it is deductable that the cousin has some form of relation (Your theory about her wishing to become a shinigami is possible) and that the murder of the twin's mother is also a clue.

As I see it, it is important to have a timeline:

15 years ago: death of Aida Gou's cousin. Last request unknown. Aida Tomoe gets severe injuries, including leg incapacity and blindness.
13 years ago: death of Shiroha's and Kureha's mother. Planed by their father, Ichirou Houjou, alongside "Minamida".
10 years ago: Shinigami Saaya completes training in 6 months. First mission, Destiny: Dead of Minamida Azuma. Coma of Minamida Azuma. Last request: "Keep watch over whether or not I die." At this part, Minamida's younger brother was stuck in bed, with a weak constitution.
Saaya promises to "not lie" and to "never make a mistake", or she would send herself to hell.
9 months ago: murders of the Aida family. Only Aida Shizuka survives. Aida Tomoe's and Aida Gou's "last request" is not known. Aida Hibiki's last request is to "Inflict maximum punishment upon Shizuka!". Aida Shizuka's memories are erased, and a picture is taken of the scene, with him holding the murder weapon.
6 months ago - present: Aida Shizuka is sentenced to execution, but escapes. Murders of 7 people by Asami, including Shiroha. Murder of Kureha by Ichirou Houjou.
Incidents regarding Kuroneko, including him obtaining information on the Shinigami and the Shinigami-sama game, in which a girl named Sawako died. Minamida sends Shizuka the picture taken at the Aida family's murder scene. Minamida captures Shizuka.

Things I gathered so far:
Not sure whether or not Shinigami are allowed to commit murder. Chief Shinigami Ayame did say that it would be a simple task to kill Shizuka.
All of the Aida are ruled out to be the murderers.
The Houjo twins, Shiroha and Kureha, can't be the murderers.
Asami can't be the murderer, due to having a weak body at the time. Plus she didn't know Shizuka or Hibiki.
Minamida Azuma, in a coma for 10 years, can't be the murderer.
Gou's cousin, dead for 15 years, can't be the murderer.
Minamida and Ichirou Houjou, although they'd be very capable of the murders, they didn't know Shizuka, nor his family members.
Kuroneko, although capable of murder, it is mentioned at first that he didn't know Shizuka or Hibiki nor commited a murder before the Shinigami-sama game
Kuroneko's name is Murata Gorou.
Unnamed girl, referred to as "X", is a girl that is willing to kill for Minamida. Minamida says she's "one" of his pieces.
Shinigami Saaya was in charge of the Aida family's murder case. She LIKELY erased Shizuka's memories, and she granted a "last request" that COULD MEAN death for Aida.
From this and taking into account the incident in which Asami murdered 7 people, I deduce that a "last request" that can mean death is not punishable untill it causes said death. Either that or the Chief Shinigami Ayame is more involved than what can be seen.
Saaya is holding the secret to what happened at the Aida family's murder case. Something "she can't say."
Although stated as "ability unknown", Satsuki has the ability to erase memories, same as Saaya.
I haven't seen anywhere that says that a shinigami's prisoner is prohibited from murder. In fact, it is mentioned that prisoners can get involved in matters in which the Shinigami can't, precisely because they are not shinigami's.
Aida Gou said "Stop it Shizuka!".
Chief Shinigami Ayame can't access Shizuka's files.

Shinigami rules:
Shinigami are to preserve "order."
Contradictions are a "sin" for shinigami.
Shinigami must complete a "last request" to their maximum capabilities.
A "last request" can't grant "murder" or "resurrection".

Clue I'm missing: everything Saaya said, quoted exactly as in the story.
Doing a skim reading, I wasn't able to find all of Saaya's words. I could only find the part where she says that if the Aida brothers came close to finding the truth, it'd be a matter of just "erasing it again". I do remember she said that she didn't know anything, but an exact quote would be better.

This is as far as I got. Up to everyone interested in finding out the truth to find any mistakes/parts I forgot, add their own found facts or their own hypothesis.