What is the meaning of the theonym "Allah" being on the left hand, and why did Allah choose the questioner to bear this name?

We do not know of any specific significance for what you mentioned. The verses of Allah, spread throughout souls and horizons, are sufficient without this. We advise you to devote yourself to the worship of your Lord and to occupy yourself with what benefits you in your religion and worldly life, both in knowledge and action. Be diligent in acts of obedience and avoid sins, for these are what will benefit a servant in the Hereafter. Do not preoccupy yourself with this matter; rather, reflect upon the Qur'an, understand it, and ponder over Allah's signs in the heavens, the earth, and in yourselves, which indicate His immense power and wisdom.