Reincarnation is not an orthodox Christian idea. Resurrection is.1
The most long standing articulation of this belief among Christians can be found in the Nicene Creed:
I look for the resurrection of the dead, and the life of the world to come.
Nicene Christians (includes all mainstream traditions, excludes some sects) believe that we live once, die once, then are resurrected either to Heaven or Hell. Although there is the idea of a life different from the one we are living now, this eternal existence is quite different from the idea of living life again in the same or similar earthly context that is 'reincarnation'.
Hebrews 9:27 (ESV)
27 And just as it is appointed for man to die once, and after that comes judgment,
Outside of established historical Christianity the lines get a bit blurry. There are few topics (the nature of Jesus Christ aside) that have generated so many aberrant teachings as the nature of the resurrection of humans. There have been many direct challenges to this (e.g. Annihilationism's belief that some or all beings simply cease to exist) and even comprehensive variant systems (e.g. Mormonism's belief in other worlds and levels of being that can be attained) but I think it is more likely that your friend subscribes to some sort of post modern conflation of religions. There are a large number of folks using the name "Christian" who do not subscribe to it's historical tenets. Most commonly they reject the divinity of Christ and/or the inerrancy of Scripture. Seeing Jesus as only a good moral teacher rather than the Christ, there is no reason not to also pick and choose various doctrines from the worlds other popular religious systems. Enter karma, reincarnation, etc.
Scripture itself contains no support whatsoever for the idea of reincarnation. However in order to come to this conclusion about it you must first agree that it means something. If not just parts but all of it is to be considered true and words are believed to have meanings and the intent of the original authors is to be considered, nothing will support your friends case. However without those a priori assumptions, of course any number of small phrases may be plucked from their original contexts and presented in support of any number of philosophical ideas.
1 See this excellent answer on a related question: What's the difference between resurrection and reincarnation?