The root of the word in Sanskrit is clear from saM (together) and dhA (establish or hold), both words of indoeuropean antiquity, the root san-dhA means to unite. This gives rise to various words with meanings as diverse as alliance, search, and evening (joining of day and night), for example.
The particular meaning of sandhini from which the name is derived, however, I do not know. As a word in Sanskrit, it can mean a female minister of alliances, or a cow who had taken a bull (and whose milk was prohibited food), or a cow that had been milked too often. On the other hand, the name might be a modern coinage from some other word in the cluster of meanings of san-dhA
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