Once you've finished writing your name, also write the date.
Literal
Name [subject-が] write-finish [conditional-たら] date [also-も] write-keep.
が instead of を marks 名前 — treating the name as the subject of the intransitive-like 書き終わる construction. The sentence reads as if 'the name finishes being written' rather than 'you finish writing the name.' A subtle aspect variant from the more typical を-marked form.