Joshua Davey at Letters from Babylon argues that free will and eternal security are inconsistent:
If this is true, then I believe the Arminian who holds to eternal security puts himself in a very difficult position indeed. For if he believes that humans exercise genuine free will in the choice whether or not to accept the salvation that Christ offers, but at the same time believes that the believer cannot lose his salvation, he is essentially saying that the believer, in accepting Christ, forfeits at least an element of his free will—he cannot “un-choose” salvation. Thus, the Arminian who holds to eternal security seems to be saying that (1) part of what it means to be made in the image of God is to have genuine free will, (2) humans exercise this genuine free will in their choice whether or not to accept the salvation offered by Christ (3) once humans have accepted Christ, they cannot exercise their free will to un-choose salvation. To me, what seems to follow from this is that the Arminian who holds to eternal security must also believe that in making the choice to accept salvation, humans surrender their free will as to that most important decision, which in essence means they surrender part of the imago dei. And because I do not believe that humans can surrender part of the imago dei, I believe that holding to both Arminian free will and Calvinist eternal security is incoherent.
I'll admit that I do not find this argument very compelling. As surrender to God is in fact a large part of what it means to accept His grace, giving up a portion of your free will does not seem incompatible with the view that it is initially your will to give. He likens making the choice to accept God to crossing a street--sure you can cross it, but you can also walk back. I tend to think of our relationship with God in more binding terms, such as a contract or, more fittingly, a marriage. A marriage cannot be easily undone, at least not as Jesus saw it (Matthew 5:31-32), where, sure, you can have a divorce, but that's just a legal fiction, not really an end to the marriage. Admittedly, even from Jesus's view, a marriage could be undone (by adultery), but it still wasn't as easily unmade as made.
Of course, I have some peculiar views on God's relationship to time and thus his relationship with us linear creatures, so I tend to view these things in a different light entirely.