The phrase in italics is from Ps 118:21 (117:21 in the LXX numbering). The psalmist says "I thank you because you heard me, and because you have become my savior/salvation." Remember that the Psalms served Israel as both a song book and a prayer book. Even a barely educated (but pious) first-century Jew would be able to sing most the Psalms from memory.
Why does Jesus refer to Ps 118 while praying? First, this Psalm is a central part of Jesus' self-understanding. The following verse, "The stone which the builders rejected has become the chief corner stone" (Ps 118:22) is quoted several times in the New Testament to describe how Jesus was rejected by the leaders of Israel but chosen by God (Matt 21:42-44, Acts 4:10-11, 1 Pet 2:4-7). Just a few weeks after the raising of Lazarus, the crowds at the Triumphal Entry acclaim Jesus as king (and therefore Messiah) by singing Ps 118:25-26 to him, "Hosanna! (Save!) Blessed is he who comes in the name of the Lord" (John 12:12-13).
Second (and perhaps most importantly), Ps 118 is an appropriate one to the situation of the death of Lazarus. It describes God's help in a time of severe need. V. 17 is especially appropriate: "I will not die, but live, and recount the deeds of the Lord."
Finally, there are three interesting possible plays on words in Ps 118. When the crowds shout "Hosanna," they are intentionally connecting the Psalm to Jesus. The Hebrew phrase is hoshiyah-nah, (save, please!), and hoshiyah is from the same root as Jesus (Yeshua = Savior, salvation). While that may be obscure to us, it is fairly clear in Hebrew. In the same way, Ps 118:21 reads "you have become my Yeshua (= Savior/salvation)." Last (and this is quite speculative, as if the rest of this post were not!), Lazarus is a nickname for Eleazar, "God has helped" - and this phrase is found in Ps 118:13.
What I find most interesting is that in a moment of deep emotion, Jesus naturally turns to a prayer that he has probably often prayed before, a prayer that communicates his own and Lazarus' deep needs. Jesus wants God to help (Eleazar), he wants God to bring salvation (Yeshua). He wants this for the sake of Lazarus' family, but mainly so that Lazarus will recount the deeds of the Lord (as he does, John 12:10-11, 17) and so that many will believe that Jesus is sent (John 11:42), or rather that he "comes in the name of the Lord" (John 12:13/Ps 118:26)
Nerd note: John uses updated Greek in this allusion to Psalm 118, as he does in other allusions to the OT. Psalm 118 uses the older epakouo for hear and exomologeomai for thank. John updates to the more normal 1st century akouo for hear and eucharisteo for thank.
The picture: The Raising of Lazarus, by Giotto di Bondone, 1308.