Though Punjabi is derived from Sanskrit, the intense interest in names and folk etymology surrounding a lot of them, and my unfamiliarity with the Punjabi language makes this set of names difficult for me. I can, however, guess: so please take the following with a huge lump of salt.
There is a Sanskrit root bhaj which originally meant to divide or to share (and is etymologically linked to beech in English: cognates in many other Indoeuropean languages are known). The sense shifted quite soon to that of bestowing, and in the middle voice, of accepting. In today's language only the sense to pray has survived, and in many languages it is a common term for singing the glory of God.
The particle ni (cognate with the begining of English nether) meant down, back, or within, and gave rise to nitya meaning innate. The meaning of this one shifted to eternal or continual, and in many languages it means daily with a strong sense of necessary and perpetual. This word became nIt in many modern languages.
So, my guess would be Bhajneet means one who constantly remembers God, or some slight variation of that.
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