The mother tried to nurture her son's interest in art.
Literal
Mother [topic-は] son [possessive-の] art [toward-に対する] interest [object-を] nurture [goal-ように] endeavored
に対する modifies 関心 = 'interest toward art.' ~ように努めた = 'endeavored so that (she would nurture).' 努める is more deliberate than ~ようとした — it implies sustained effort, not just an attempt.