Eckersley, released by Bolton at the end of the season, will sign a one-year deal in time for the start of the npower League Two campaign.
"I came here on trial a couple of weeks ago looking for a contract and that's what I've done. I must have impressed enough to get it," said the 20-year-old.
"There were a lot of lads here on trial at the start and I just had to believe in myself and do the best I could."
Stanley manager Paul Cook added: "Tom has impressed us in pre-season and now we're looking for him to continue improving.
"We have good competition in the centre of defence and that's a good thing for the team."