Christianity believes Jesus is God, or more specifically one person in God - the others being the Father and the Holy Spirit. The questions appears to accept that Jesus and God the Father are to be treated as separate persons in the Trinity, but asks for the biblical basis for concluding that Jesus is Yahweh, not the doctrinal basis for any such conclusion. In the following analysis I will refer to YHWH or Yahweh, but the English translation, Jehovah, is equally applicable.
In spite of common belief, we do not really know what the tetragrammaton, YHWH, means. It has always been written without vowel indicators and, since the Jews ceased saying the word after the Romans destroyed the Temple in 70 CE, we do not know how it was once pronounced and therefore what vowels were included in the name.
Othmar Keel and Christoph Uehlinger suggest in Gods, Goddesses, and Images of God: In Ancient Israel, page 393, Yahweh might originally have meant ‘he blows’, as a reference to his perceived original role as storm God. Lester L. Grabbe says in Ancient Israel, page 153, there is considerable – if not universal – support for the view that early Egyptian inscriptions mentioning Yhw, worshipped as a storm god in Midian, refer to the name Yhwh and that worship of Yhwh did not originate with Israel but was picked up sometime in the pre-settlement or settlement period. The Bible Study glossary agrees that ‘he blows’ has been suggested by scholars, based on etymology, along with 'He Is' and 'He Causes to Be', but, importantly in the present context, none of these has won general acceptance. The point here is not to prove that YHWH means something other than 'He Is', but that we do not know what it does mean, other than an ancient name for God.
If we can not be at all sure what YHWH meant in Old Testament usage, we can not speculate based on the assumption that in the Old Testament the name actually means 'He Is'.
Wikipedia says that Protestant commentaries often state that whenever John reports Jesus as saying ego eimi ('I am'), a claim to deity is implicit. This connection is made because it is assumed that ego eimi is related to I am that I am or Hebrew Ehyeh-Asher-Ehyeh in Exodus 3:14. A claim to deity is not at all controversial for Jesus, but this does nothing to justify a conclusion that the name Yahweh is as valid for the Son as for the Father.