My personal theory is that it dates from the early MDL period, when leases were first sold on berths, exerting an upward pressure on price, gentrifying the Hamble and evicting the budget end of the yottie community to the upper reaches of the river.
With its new, more upmarket image, the Powers That Be decided that a posher name was called for, hence adding "Le Rice" to good, old-fashioned "Hamble". Rather like calling Battersea "South Chelsea" /forums/images/graemlins/grin.gif
More seriously, have a look here. To quote briefly:
"The spelling of the village name has varied considerably over the centuries, examples being Hamelea circa 730, Hammel in 1496, and Ham-en-le-Rice in 1846. Today it is officially known as Hamble-le-Rice. The derivation of the name Hamble is uncertain, but 'Hamel' is Old English for 'crooked,' which could refer to the course of the river. 'Le-Rice' means 'brushwood' or 'the rise,' which would be appropriate since the church stands 50 ft above sea level. The name, however, could come from a Saxon Thane called 'Hamele.'"
Le Rice is a mutation of Le Rise, it refers to the hill from the quay up to the village square, agreed it was just plain Hamble in the 1970's, the recent pretentious addition of the old name is a bit tacky.
Aside from boating it has a lot of history as a village, plots to overthrow kings hundreds of years ago to 7 air strips in the 20th century.