Abandonment is not the legal concept to be concerned with (though the situation might fall within the ambit of a law that uses the word "abandon"), instead the question should be about the legal obligations of a parent. California Family Code is what you want to look at. Though a look at the criminal act of "child abandonment" can be informative: section 271 and following indicate that there are some penalties for abandonment-like actions for children between 14 and 18, but the acts would have to be "willful" and "without lawful excuse" (which probably includes "inability to perform").
In the Family Code, section 7822 states when proceedings can be brought. For example, if
(2) The child has been left by both parents or the sole parent in the
care and custody of another person for a period of six months without
any provision for the child's support, or without communication from
the parent or parents, with the intent on the part of the parent or
parents to abandon the child.
Section 3900-3901 says that
the father and mother of a minor child have
an equal responsibility to support their child in the manner suitable
to the child's circumstances. The duty of support imposed by Section
3900 continues as to an unmarried child who has attained the age of 18
years, is a full-time high school student, and who is not
self-supporting, until the time the child completes the 12th grade or
attains the age of 19 years, whichever occurs first. Nothing in this
section limits a parent's ability to agree to provide additional
support or the court's power to inquire whether an agreement to
provide additional support has been made.
There seems to be a formula for computing expectations of support: but the law won't require a person to pay money that they do not have. The law also will not compel a third party to take in an guest, nor will it compel the mother to become homeless (i.e. order the third party to take in the child or eject the mother). The courts could easily require the mother to take financial responsibility for the child.