Question
What is the meaning of Brahman & Vasudeva?
Answer
The definition of Brahman is given in this chapter of the Taittiriya Upanishad:
yato va imani bhutani jayante ।
yena jatani jivanti ।
yatprayantyabhisaṃviśanti ।
tadvijijñasasva ।
tad brahmeti ।That from whence these beings are born, that by which, when born, they live, that into which they enter at their death, try to know that. That is Brahman.
This is the passage that Adhyaya 1 Pada 1 Sutra 2 of the Brahma Sutras is based on:
janmādyasya yataḥ
That (is Brahman) from which (are derived) the birth etc., of this (universe).
Now concerning the name , it has two meanings. As a name of Krishna, Vāsudeva refers to the fact that he is the son of Vasudeva. But Vāsudeva has been a name of Vishnu long before the time of Krishna. For instance, this chapter of the Valmiki Ramayana refers to Vishnu's incarnation Kapila as Vāsudeva. How is this possible? It's because Vaasudeva has another meaning. As a name of Vishnu, it refers to the fact that since Vishnu is the supreme Brahman, he is the one who dwells in all beings and in whom all beings dwell. Here is what this chapter of the Vishnu Purana:
The term Vāsudeva means that all beings abide in that supreme being, and that he abides in all beings
This chapter of the Udyoga Parva of the Mahabharata says much the same thing:
He is called Vasudeva in consequence of his enveloping all creatures with the screen of illusion, or of his glorious splendour, or of his being the support and resting-place of the gods. He is called Vishnu because of his all-pervading nature.
EDIT: You asked for the etymological meaning of these terms. The meaning of Vasudeva given in the Vishnu Purana and the Mahabharata is already an etymological meaning, as should be clear from seeing the original Sanskrit. Here is the Udyoga Parva verse I quoted above:
vasanāt sarvabhūtānāṃ vasutvād deva yonitaḥ
vāsudevas tato vedyo vṛṣatvād vṛṣṇir ucyate
So the word comes from the root "Vas", which means to dwell or abide.
And the etymological meaning of Brahman is given in this section of Adi Shankaracharya's Brahma Sutra Bhashya:
Brahman, which is all-knowing and endowed with all powers, whose essential nature is eternal purity, intelligence, and freedom, exists. For if we consider the derivation of the word 'Brahman,' from the root brih, 'to be great,' we at once understand that eternal purity, and so on, belong to Brahman.
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