Citing sources it did not identify, the network said the phrase "My E-L-F weapon" was carved into the stock of the shotgun Alexis used in Monday's rampage, which ended when he was killed in an exchange of gunfire with security personnel. It wasn't clear what "E-L-F" stands for, though ABC said its law enforcement sources said there was speculation it could stand for "extremely low frequency" since Alexis allegedly had talked about hearing voices and having "vibrations sent to his body" by a " microwave machine."
Investigators said they were analyzing whether "better off this way" meant Alexis knew he would die, ABC reported.
Police were still trying Wednesday to establish a motive for the killing spree.
Alexis' mother told CNN she is heartbroken by what her son did.
"I don't know why he did what he did, and I'll never be able to ask him why," Cathleen Alexis. "I'm so, so very sorry this has happened. My heart is broken."
White House press secretary Jay Carney announced Wednesday President Barack Obama will attend a memorial service Sunday to honor the massacre victims. Carney did not provide details on the time or location of the service.
The Navy League of the United States, a non-profit that promotes a strong U.S. maritime defense, said it has taken steps to create a survivors fund for the injured victims and families of those killed.
"It is our duty, as a trusted partner of the sea services, to do what we can to ease the burden of those whose lives are forever changed by this tragic event," League President James H. Offutt said in a release. "We look forward to working with like-minded nongovernmental organizations, community leaders and others on this endeavor. We are going to do our best to help members of our extended Navy family through this difficult time."
Aaron Alexis, 34, was a defense contract worker who had been granted access to the secure Navy facility in Washington when he entered two days ago and used a shotgun -- and handguns taken from yard guards -- to kill 12 people before he was killed. Questions persist about how his deteriorating mental health status was missed but evidence suggests matters could have been much worse.
He tried to buy an assault rifle at a Virginia gun store but was refused, CBS News reported Wednesday.
The store wouldn't sell him an AR-15 after he test-fired it, though the reason for the refusal was unclear, the report said. Alexis was able to use an out-of-state ID to buy a shotgun investigators believe is the one used in the rampage.
As Washington police dealt with the situation at the Navy Yard, the men in charge of security on Capitol Hill reacted differently about how protect members of the House and Senate and their staffs, Roll Call reported.
Terrance Gainer, the Senate's sergeant-at-arms, locked down his half of the Capitol and Senate offices. House Sergeant-at-Arms Paul Irving did not.
About an hour after the shooting at the Navy Yard began Monday, Gainer sent a Twitter message saying there was "no known direct threat" to the Capitol, but Capitol police were "taking extra precautions."
At 1:30 p.m., Irving sent an email to House staff saying his office was "evaluating available intelligence and evidence," but that no one at the Capitol appeared to be at risk.
Both chambers convened briefly at 2 p.m., and then adjourned about 10 minutes later.
Shortly after 3 p.m., Gainer ordered a lockdown of the Senate that was partially lifted at 4:18 p.m., which allowed people to go home but no one to enter.
While saying he respected Irving's decision not to lock down the House, Gainer said it would have been "more effective and less troublesome" if both chambers had acted in concert.
The result, he said, had limited effectiveness, but "served its purpose."
Police in Newport, R.I., told the Navy a month ago Alexis reported hearing voices. The incident and others point to shortcomings in a system in which Alexis' behavior fell below a level that would have brought a serious response from authorities, experts and officials cited by The New York Times said.
Alexis worked on a computer project as a contractor at the Navy Yard for several days before the shooting, The Washington Post reported.
He used his secret-level clearance to gain entry to the former shipyard and ordnance plant Monday.
He was granted secret-level security clearance in March 2008, when he was working as a full-time Navy reservist, the Pentagon said.
A Defense Department official told the Post the type of clearance Alexis had was typically good for 10 years.
Former military employees who become private-sector contractors can maintain their clearance, the official said, explaining a person can keep the clearance "in the absence of unadjudicated derogatory information."
Alexis was discharged from the Navy in January 2011 after a series of run-ins with his military superiors and police.
While investigators learned more about Alexis, Obama ordered the White House budget office to conduct the review of policies for security clearances for contractors and employees in federal agencies.
The president told Spanish-language TV network Telemundo the nation did not have a "firm-enough background-check system."
Defense Secretary Chuck Hagel also ordered a broad review into the physical security and access to military installations worldwide.
Obama repeated his call for Congress to enact legislation to keep guns out of the hands of criminals and those with mental illness.
"I do get concerned that this becomes a ritual that we go through every three, four months, where we have these horrific mass shootings," he told Telemundo.
Congressional leaders had no immediate response to his gun-control call.
Several senators called for a congressional investigation into the granting of security clearances to government contractors.