Both names come directly from the Finnish word for “blue.”
The name is borne by Swedish Social Democrat politician, Sinikka Bohlin (b. 1947) as well as by Norwegian vocalist Sinikka Langland.
The designated name day in Finland is September 2nd.
To hear what both names sound like when pronounced by native speakers, consult this link: http://www.forvo.com/search/Sinikka/ and for Sini http://www.forvo.com/word/sini/