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Not extremely explanatory, but this from WordOrigins.com:

This word meaning a rolling platform on casters is American in origin, dating to about 1901. From Samuel Merwin's and Henry Webster's Calumet “K” of that year:

Other gangs were carrying them away and piling them on “dollies” to be pulled along the plank runways to the hoist.

Dolly is used prior to this to refer to several different devices and tools, however. The first of these was a washtub agitator that had four extensions and resembled a doll with arms and legs. From William Roberts's 1792 The Looker-On:

The Dumb Dolly, or a machine for washing, is recommended.

Since then, dolly has been applied to various devices, few of them resembling dolls.