I've heard some people claim that spinal marrow or spinal cord is haram to eat. Is this claim true and what is the basis for this?
No, it is not haram. At all. The prohibition is against blood, and marrow is not blood. Some may have confused that issue.
Everything is Halal until proven otherwise. Allah said, “It is He Who hath created for you all things that are on earth.” (Quran 2:29). In other words, spinal marrow or spinal cord is Halal.
Majority of the Muslim scholars hold of its permissibility. However, some from the Shia religion rely on a fabricated hadith, "The worst of my ummah are those who consume bone marrow" (Bihar al-Anwaar , 62/293).
All in all, there is no evidence of prohibition.