House Intern Pay Act of 2019 This bill addresses the compensation of interns for Members of the House of Representatives. The bill modifies a capped allowance for compensating interns who serve in the offices of Members of the House during a calendar year. The Committee on House Administration must establish an equivalent allowance for each Member's office for that purpose. An unused portion of the allowance may be reallocated to another Member's office, depending on specified circumstances. Furthermore, the bill requires that the hourly rate for interns be at least $15 per hour for the first year and provides for inflationary adjustments in subsequent years.