。
She asked me to come to her bedside.
Literal
She [topic-は] me [object-を] bedside [to-に] come [indirect command-ようにと] asked.
The pattern ~ようにと頼む is the standard frame for indirect commands and requests — '[someone] asked [someone] to do X.' よう nominalizes the requested action; と marks it as quoted content; 頼む supplies the asking. Within this construction, the addressee (私) takes を — the standard Japanese frame is Xを + [clause]ようにと頼む for 'ask X to do something,' even though plain 頼む elsewhere often takes the addressee with に. 枕元 ('bedside,' literally 'pillow-side') often carries the connotation of attending to someone ill or dying — a quietly weighty image.