We find no effects on any educational outcomes, including grades, test scores, credits earned, attendance, and disciplinary actions

From Will a Computer at Home Help My Children in School? by Timothy Taylor. Too small a study (N=1,100) and too small a duration (one or two years) to be anything other than indicative. The conclusions?

"Although computer ownership and use increased substantially, we find no effects on any educational outcomes, including grades, test scores, credits earned, attendance, and disciplinary actions. Our estimates are precise enough to rule out even modestly-sized positive or negative impacts. The estimated null effect is consistent with survey evidence showing no change in homework time or other “intermediate” inputs in education."

Taylor comments:

Having a computer at home increased computer use. Students without a computer at home (the "control group") reported using a computer (at school, the library, or a friend's house) about 4.2 hours per week, while students who now had a computer at home (the "treatment group") used a computer 6.7 hours per week. Of that extra computer time , "Children spend an additional 0.8 hours on schoolwork, 0.8 hours per week on games, and 0.6 hours on social networking."

Of course, any individual study is never the final say. Perhaps having access to a home computer for several years, rather than just one year, would improve outcomes. Perhaps in the future, computer-linked pedagogy will improve in a way where having a computer at home makes a demonstrable difference to education outcomes. Perhaps there is some overall benefit from familiarity with computers that pays off in the long run, even if not captured in any of outcomes measured here. It's important to remember that this study is not about use of computers in the classroom or in education overall, just about access to computers at home.

My suspicion is that, as is normal, it is not access to computers per se but the behaviors and values you bring to the task of studying that are the determinant of outcomes.