General Erinmore chose him on purpose because of the fact that Blake's brother was in the same regiment who was going to be pushed forward.
Blake will be motivated to somehow get to his brother's regiment and deliver message to save them all.
Following are dialogues from tent briefings at beginning of movie.
SERGEANT SANDERS: Lance Corporals Blake and
GENERAL ERINMORE: Which one of you is Blake?
ERINMORE: You have a brother, a Lieutenant in the 2nd Devons?
BLAKE: Yes, sir. Joseph Blake. Is he-
ERINMORE: Alive, as far as I know. And with
your help I’d like to keep it that
Since the communication lines were cut Erinmore wanted someone with map skills.
ERINMORE: Sanders tells me you’re good with
maps. That true?
BLAKE: Good enough, Sir.
It was a smart thinking. An unmotivated soldier might desert in middle of a mission due to the risk associated with it.
Update after Greek - Area 51 Proposal's comment: Thanks, but why didn't Gen Erinmore include a junior commissioned officer with Blake?
Also, Blake and Schofield carried written command by General Erinmore which was supposed to be obeyed by Colonel Mackenzie. Delivering it by a private or by a captain would make no difference.