The Rationale behind NIST's counter/LFSR recommendation on deterministic IV constructions on SP 800-38D
I can't tell whether this is what motivated NIST, but I found a paper that has an argument for LFSR counters in some applications:

Mukhopadhyay, Sourav and Palash Sarkar. 2006. &quot;Application of LFSRs for Parallel Sequence Generation in Cryptologic Algorithms.&quot; Cryptology ePrint Archive: Report 2006/042.


Abstract. We consider the problem of efficiently generating sequences in hardware for use in certain cryptographic algorithms. The conventional method of doing this is to use a counter. We show that sequences generated by linear feedback shift registers (LFSRs) can be tailored to suit the appropriate algorithms. For hardware implementation, this reduces both time and chip area. As a result, we are able to suggest improvements to the design of DES Cracker built by the Electronic Frontier Foundation in 1998; provide an efficient strategy for generating start points in time-memory trade/off attacks; and present an improved parallel hardware implementation of a variant of the counter mode of operation of a block cipher.

As I mention in one of the comments to the question, NIST SP 800-38a (&quot;Recommendation for Block Cipher Modes of Operation,&quot; 2001 Edition) also mentions LFSRs as an example of an allowable incrementing function for counter blocks (p. 18), so the technique predates both GCM and this paper.
RFC 3686 (&quot;Using Advanced Encryption Standard (AES) Counter Mode With IPsec Encapsulating Security Payload (ESP),&quot; January 2004) also allows the use of LFSR counter blocks. It says LFSR are a common IV generation method (p. 5) and obliquely appeal to similar motivations as Mukhopadhyay &amp; Sarkar for allowing them:

[p. 14] Allows adders, LFSRs, and any other technique that meets the time      budget of the encryptor, so long as the technique results in a unique value for each packet.  Adders are simple and straightforward to implement, but due to carries, they do not execute in constant time.  LFSRs offer an alternative that executes in constant time.

Saying that LFSRs execute in constant time does counterintuitive to us software folks, who are biased into thinking more generally that addition is &quot;simple&quot; but LFSRs &quot;complex.&quot;