Geethika is from the Sinhala / Sanskrit Gītika (गीतिका — short song, hymn) — diminutive of Gīta. Geethika is one of the Sinhala + Hindi feminine names — central to traditional Sri Lankan + Indian Hindu + Buddhist heritage. The Sanskrit Gītā (song) refers to the Bhagavad Gītā (Song of the Lord, c. 5th-2nd c.
Geethika does not currently appear in the US Social Security Administration's top 1,000 girls' names, so we don't publish a US rank or birth count for it. That says nothing about the name's standing elsewhere in the world — only that it sits outside the ranked US data we rely on.
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In Pythagorean numerology the letters of Geethika reduce to 3, The Communicator. This is a traditional interpretive system, not a factual claim about the name.