In Rev 3:5, we read the following:
He that overcometh, the same shall be clothed in white raiment, and I will not blot out his name out of the book of life, but I will confess his name before my Father, and before his angels.
This verse is a very strong text against the various theologies of "eternal security," as it shows that the names of those contained in the book of life (i.e., those who will be saved) are not static, but that one's name can be "blotted out" (εξαλιφω [alt. "to erase"]) therefrom (i.e., the loss of one's justification). Contingent upon remaining in the book of life (read: in a saved state), one must persevere to the end (cf. Phil 2:12), or, as the Greek of this verse reads, one must be ο νικων ("the conquering one").
The Reality is Not Optional youtube page, one that supports Open Theism, has a great 20-min presentation refuting Perseverance of the Saints/Eternal Security on the issue of names in the Book of Life in the book of Revelation, including Rev 3:5: