。
She can't even write her own name.
Literal
She [topic-は] her own name [even-すら] write [thing-こと-が] able-not.
すら is a focus particle meaning '(even) X,' more literary and emphatic than the everyday さえ. Stacked onto 自分の名前 it underlines how basic the missed skill is — illiteracy so deep that even her own name is beyond her. 出来ない is the negative potential of する, with 事ができない as the formal/written potential construction. Note 事 is written here in kanji — modern usage often prefers kana for the grammatical こと in this construction.