My son has scholarship with out-of-state university.Since scholarship funds for room & board and are considered as taxable income, does he have to pay taxes in that state

Excess scholarships are considered taxable income and unearned kiddie taxable so must be reported if kiddie unearned income is in excess of $1,050 along with other unearned income.  This is taxable income but it is not "earned income" so it is not subject to social security and medicare tax, just state and federal income tax.  You are the first person to ask if it was taxable to the non-resident state.  I am asking the question to other TT support to see if anyone else thinks otherwise.

If you earned more than $1,000 in Oklahoma and are a nonresident, then you are required to file a nonresident return in Oklahoma.  

The amount is not considered earned income though.   It may only be taxable to your resident state based on the definitions given.

OK nonresidents sourced income per OK is:

The sources of income taxable to a nonresident are:
(1) Salaries, wages and commissions for work performed
in Oklahoma.
(2) Income from an unincorporated business, profession,
enterprise or other activity as the result of
work done, services rendered, or other business
activities conducted in Oklahoma.*
(3) Distributive share of the Oklahoma part of partnership
income, gains, losses or deductions.*
(4) Distributive share from Sub-chapter S Corporations
doing business in Oklahoma.*
(5) Net rents and royalties from real and tangible personal
property located in Oklahoma.
(6) Gains from the sales or exchanges of real and
tangible personal property located in Oklahoma.
(7) Income received from all sources of wagering,
games of chance or any other winnings from
sources within this state. Proceeds which are not
money shall be taken into account at fair market
value.