First, I understand that Voldemort wanted to use the Stone to restore him to normal life. But what I don't get is - why does he want to use the Stone for that?
He had a wizard servant at his disposal, like he had with Pettigrew later, and he had his Horcruxes in place. Why did he bother with the Stone? He could just have made Quirrell perform Pettigrew's job without the need to move to Hogwarts, which would eliminate all the risks of being discovered that would bring.
Since Quirrell met Voldemort on a journey, he could just have never returned from that journey and most (with the possible exception of Dumbledore) might just suspect some tragic accident; so that way of resurrection could be performed in complete secrecy - while dwelling in Hogwarts, with hundreds of wizards and witches (most underage, but including several skilled teachers and Dumbledore) seems much more risky to me. Why did Voldemort take that risk?
Two unconvincing ideas I had:
The reason could be that there was no way to get hands on Harry. However, as Voldemort mentions to his Death Eaters after his resurrection, he could have used any wizard or witch that still hated him. True, he would not have the special effects of Harry's blood, but that's also true if he had used the elixir.
Quirrell might be unable to perform the ritual. That seems unlikely, since Quirrell isn't depicted as a great wizard, but neither is Pettigrew, who's an awful wizard and could do the job.