Do tRNAs that recognize multiple codons have any preference for one over another?
There is an extensive literature on synonymous codons and their usage in different organisms. I shall assume that the question concerns relationship between strength of codon–anticodon interaction and the efficiency (speed in this case) of translation.


It is not disputed (I am aware of few experiments on this) that the strength of codon–anticodon interaction depends on the number of hydrogen bonds made.
It was predicted that for optimal rate of translation one needs a balance between association and dissociation of the codon–anticodon pair. Hence weak interactions (all AU) — which would disfavour association — or strong interactions (all GC)  — which would disfavour dissociation — would not allow a rate of protein synthesis as fast as that for intermediate interactions (mixed AU,GC).
This appeared to be borne out 
in the codon usage patterns of rapidly growing organisms such as Escherichia coli, something subsequently confirmed by more extensive studies. To be precise, the more highly expressed proteins (e.g. ribosomal proteins) were encoded by mRNAs containing the intermediate  codon–anticodon interactions predicted to allow rapid translation. In contrast, poorly expressed proteins (e.g. repressors) were encoded my mRNAs predicted to produce slower translation.
One should be aware, however, that there are other factors that influence the codon usage in different organisms, and that these can overide the effect of codon–anticodon interaction strength.