Krishna comes from the word Akarshan. Karshati iti Krishnah: “The one who attracts is Krishna.” So Krishna can attract everybody. Who else but God can be all attractive? Krishna is all-attractive. Therefore he is God!
Once an interviewer asked His Divine Grace Srila Prabhupada, “But what about a name
meaning "all-powerful"?”
To this he replied, “Unless you are powerful, how can you be all-attractive?”
We must understand that the name ‘Krishna’ includes everything. So if any name be given to God, it must be Krishna! Whenever we see any beautiful thing in this world, we should understand that it is nothing but a spark of the splendorous beauty that Krishna possesses.
arjuna uvāca
paraṁ brahma paraṁ dhāma
pavitraṁ paramaṁ bhavān
puruṣaṁ śāśvataṁ divyam
ādi-devam ajaṁ vibhum
(Bhagavad Gita 10.12)
"Arjuna said: You are the Supreme Brahman, the ultimate, the supreme abode and purifier, the Absolute Truth and the eternal divine person. You are the primal God, transcendental and original, and You are the unborn and all-pervading beauty."
Arjun after hearing from Krishna became completely free from all doubts and accepted Krishna as the Supreme Personality of Godhead. Some might argue, however, that Arjun's statement was motivated by his personal affection for Krishna or perhaps even fear of Him. But Arjun cleared any such possibility by mentioning all such great authorities as follows:
āhus tvām ṛṣayaḥ sarve
devarṣir nāradas tathā
asito devalo vyāsaḥ
svayaṁ caiva bravīṣi me
(Bhagavad Gita 10.13)
"All the great sages such as Narada, Asita, Devala, and Vyasa proclaim this of You, and now You Yourself are declaring it to me."
Krishna himself announces this:
mattaḥ parataraṁ nānyat
kiñcid asti dhanañ-jaya
mayi sarvam idaṁ protaṁ
sūtre maṇi-gaṇā iva
(Bhagavad Gita 7.7)
"O conquerer of wealth [Arjuna], there is no Truth superior to Me. Everything rests upon Me, as pearls are strung on a thread."
So we must accept Krishna as the Supreme Personality of Godhead on the authority of such scriptural evidence.