Is one allowed to blow their nose in the middle of Shemoneh Esrei?
It is not only permitted, but also recommended that one who feels the need should blow his nose during Shemoneh Esrei.
The reason for this is that needing to blow one’s nose, without being able to do so, is distracting (similar to having an itch, or needing to use the bathroom), and will prevent a person from having proper intent during his davening.
The situation might also cause the person to wipe his nose with his hand intermittantly, which is disrespectful to the prayer.
It is therefore correct to momentarily stop davnening, blow one’s nose, and then continue with a clear head.