Wilson Follett, Modern American Usage (1966) vigorously opposes applying a possessive 's to an inanimate thing (as in "this blog's existence"), calling such possessives "false" and concluding that "we must stick to the ancestral rule which, with a few exceptions, reserves possessives in 's for ownership by a person." Shoe's answer cites a similar (though more cautiously worded) conclusion by Ron Cowan in The Teacher's Grammar of English With Answers (2008).
Living (as Follett did) in a country whose national anthem includes "the dawn's early light" and "the twilight's last gleaming" in its first two lines and "the rocket's red glare" three lines later, I find it hard to take this asserted necessity (or preference) seriously. And since "The Star Spangled Banner" was written in 1814, it appear that writers have been ignoring the claimed preferable treatment of inanimate possessives for a long time.
Bryan Garner, Garner's Modern American Usage (2003) presents what I take to be a much more realistic view of the subject in a multipart discussion of possessives:
H. Inanimate Things. Possessives of noun denoting inanimate objects are generally unobjectionable. Indeed, they allow writers to avoid awkward uses of of—e.g.: the book's title, the article's main point, the system's hub, the envelope's contents, the car's sticker price. [Cross reference omitted.]
The old line was that it's better to use an "of phrase rather than the 's to indicate possession when the possessor is an inanimate object. Write foot of the bed, not the bed's foot." Robert C. Whitford & James R. Foster, Concise Dictionary of American Grammar and Usage 96 (1955). Foot of the bed, of course is a SET PHRASE, so the example is not a fair one. As a general principle, though, whenever it's not a violation of idiom, the possessive in 's is preferable {the hotel's front entrance} {the earth's surface}.
But such possessives can be overdone. For example, avoid using the possessive form of a year—e.g.: "Mr. Rogers, 41, took the show by storm in 1993, winning 28 blue ribbons and the Show Sweepstakes with a total of 1,120 points (which really upped the ante: 1992's winner [read the 1992 winner] scored only 387 points)." Anne Raver, "A Big Flower Show One Loves to Hate," N.Y. Times, 2 Mar. 1995, at B5.
Follett's argument is premised on the notion that, inherently, some "of" phrases are not properly convertible to 's form:
The truth is that these [previously listed] possessive's in the 's form are newfangled and false. The error in writing them is to assume that Florida's governor means the same thing the governor of Florida. At that rate the Book of Revelations would be the same as Revelations' Book. The of in these phrases is not a true possessive but a defining and partitive of, as in loss of breath, piece of wood, ruler of men.
I find this argument unpersuasive. In the first place, if we accept the validity of the assertion "Florida has a governor," it seems entirely reasonable to ask "Who is its governor," rather than "Who is the governor of it?"—and by extension to ask "Who is Florida's governor?" (if we want to) rather than "Who is the governor of Florida?"
Indeed, if inanimate objects are not to be granted 's possessives, by what rationale do we freely use its in connection with inanimate objects? If we're willing to concede a relationship of possessor and possessed between an inanimate object and either an animate one (as in "the customers of the bank") or an inanimate one (as is "the holdings of the bank"), I don't see why expressing that possession with 's should be a problem for scrupulous writers.
I also wonder whether "false possessives" that use "defining and partitive of" (Follett's wording) wouldn't be more usefully considered in terms of their being what Garner calls "set phrases." Certainly the fact that God qualifies as animate doesn't prevent the conversion of "a man of God" into "a God's man" from significantly altering the original sense of the phrase. The would-be possessor's status as animate (or not) thus appears not to have much to do with whether any particular conversion of "X of Y" to "Y's X" survives that conversion with its original meaning intact.
Garner's conclusion that "As a general principle, though, whenever it's not a violation of idiom, the possessive in 's is preferable" goes farther than I think is necessary. To me, the crucial question is this: When you convert "X of Y" to "Y's X," does the meaning change or become unidiomatic? In the case of "the existence of this blog" and "this blog's existence," I think that the meaning does not change substantially, so I see no reason to avoid using the latter form if that's what you prefer.