Richard Dawkins proposed calling the phenotype a "vehicle". David Hull proposed using the term "interactor". Gouyon, Henry and Arnould proposed the term "gene avatars".
The Dawkins "vehicle" conception has a curious history. Dawkins at first promoted it. When faced with Hull's near synonym "interactor" he said (1982):
Much the same point has been realized by Hull. (1981), but I prefer to persist in expressing it in my way rather than to adopt his terminology of "interactors" and "evolvors".
Later in "Burying the Vehicle" Dawkins wrote:
I coined the vehicle not to praise it but to bury it. This is, paradoxically, why vehicle is a better name than Hull’s interactor. Interactor comes too close to the (messy) truth and therefore doesn’t merit a helpfully decisive burial.
Dawkins subsequently went on to use the concept as though it had not been "buried" at all, for example here:
The key is to abandon the word 'fitness', replacing it with the concepts of replicators, vehicles and the extended phenotype.
These contradictory approaches to the "vehicle" term have become part of the terminology's baggage.
I find "interactor" a bit of a mouthful, while "vehicle" seems somewhat more acceptable. Hull's various criticism of the term "vehicle" seem bogus to me. For example, Hull claims that the term "vehicle" makes the organism seem too passive - like a giant robot controlled by its genes. I think Hull must have not used public transport very much. I also classify Hull's claimed distinctions between "interactor" and "vehicle" as being "mostly fluff".
Both "interactor" and "vehicle" are historically tied in a bit with the "replicator" concept - which has received its share of criticism. In particular, the etymology of "replica" strongly suggests high-fidelity copying - while in most of evolutionary theory, copying fidelity is a parameter, which can take a wide range of values. For some, this appears to have tainted "interactor" and "vehicle" by association.
"Phenotype" seems to have remained the most popular term. It has useful cousins: phene, and phenome. The terminology does have some disadvantages, though: The term "phenome" has a very muddled definition - which (confusingly) fails to mirror the definition of "genome". The terms "phenotype" and "genotype" are (confusingly) used to play multiple roles - referring to both all the traits in an organism and to a particular trait.
Like the term "evolution", the term "phenotype" doesn't come in "meme" and "gene" varieties. Just as it would be nice to have access to the terms "memevolution" and "genevolution", it would be nice to have "meme" and "gene" variants of the term "phenotype". Memeticists have proposed "phemotype" has been proposed as a term applicable to cultural evolution - but this term seems to me to be too irregular to endorse.
The Genotype-phenotype distinction neatly foreshadows the rise of information theoretic accounts of evolutionary processes. The idea is a lot like the hardware/software distinction in computer science.
For some reason, the Genotype-phenotype distinction has caused much controversy in cultural evolution. It all seems tied in to the confusion surrounding cultural genes: memes. I'm not really sure why this confusion exists - these all seem like basic, foundational concepts to me.