Translation:She was accustomed to write to me once a week.
The accusative here is used to indicate time/measurement. Examples:
Mi venos lundon = I will arrive on Monday. →accusative indicates a moment in time rather than just the day ‘Monday’.
Ŝi faris tion tri fojojn = She did that thrice. →accusative indicates that the thing she did happened three times; i.e., it is a measurement.
Ŝi sukcesis la trian provon = She succeeded on the third try. →accusative indicates a point in time: when she tried for the third time, she succeeded.
La tablo estas unu metron longa = The table is one meter long. →accusative indicates the length of the table, i.e., it is a measurement. Had we forgone the accusative, then we would get the very odd sentence: La tablo estas unu metro longa = The table is a long meter. (Because metro longa = longa metro = a long meter.)
I hope this helps!