بسم الله and الحمد لله . The word يَوْم literally means the day i.e. from sunrise to sunset. This is not the intended meaning in this verse. What is meant here is a period of time i.e. the time of the Day of Judgement.
The word الدين literally means reward or reckoning. The word الدين comes from the root word دان يدين to reward or to pay back. The verb is used in the following Arabic phrase دِنْتُه بِما صَنَعَ I rewarded him for what he did. Another example of its use is the following narration : الكَيِّسُ مَنْ دَانَ نَفْسَهُ وَعَمِلَ لِمَا بَعْدَ الْمَوْتِ The wise is the one who takes himself to account and works for the Hereafter.
If we put this together we can now see why Allah named the Last Day as يوم الدين The Day of Accounting, as this is the day when everyone gets repaid for their actions in this world. The day is like a mega pay day, our deeds will be counted and accounted for. The total pay that is due will be given to us, except the reward is not money. Instead the reward on that day is the pleasure or anger of Allah. May Allah save us from His anger and may He be pleased with us on يوم الدين The Day of Accounting.
References
التفسير الوسيط للقرآن الكريم ، محمد سيد طنطاوي
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