Tarcila is a misspelling of Tarsila, a Catalan and Spanish name. In the case of your grandmother, being from Majorca, the name is Catalan (moreover, in European Spanish, ci and si are not mixed up since they sound differently, but in Catalan they sound the same, then the misspellings between c+e/i and s+e/i are very usual).
Tarsila comes from the personal medieval Latin name Tharsilla, from a dark origin. It could derive from the greek thársos, "courage, bravery", with the diminutive suffix -illa or from the personal medieval Latin name Tharsicius, with a change of suffix.
The name Tarsila was popularised by saint Tarsila, saint Gregory Magnus's aunt, a Roman virgin from 4th c. (name day Dec 24).
The name Tarsila is pronounced [tər'silə] in Catalan (tuhr-SEE-luh) and [tar'sila] in Spanish (tahr-SEE-lah).