Our daughter was a full time student and worked. She did not get her stimulus pay. Should we claim her (and her not get that pay) or should we not claim her so she will?

You do not have to claim her, but she still cannot "claim herself" if she is eligible to be your dependent.

If a child qualifies to be claimed as a dependent by someone else, they must answer "yes" they can be claimed, even if the person who could claim them agrees not to, and that will disqualify them from the rebate credit.  Answering "no" to get the rebate when they should answer "yes" is potentially tax fraud. 

Whether a college student-child can be claimed as a dependent requires an analysis of their living arrangements, income, and other issues.  There are two ways a person can qualify as a dependent, called "qualifying child" and "qualifying relative."

A child can't be a "qualifying relative" dependent if they have more than $4300 of gross taxable income, so that's probably easy to eliminate.

For your own child who is under age 24 and a full time student, the key questions will be,

a. does the child provide more than half their own support

b. does the child live with you (the parents) more than half the year.

For the "live with" question, being away at college is considered a temporary absence and the child is still considered to live "at home", unless they have actually taken firm and definitive steps to move out--like getting an apartment they pay for themselves.  If they still come home over summers and most of their stuff is still at your home they probably still "live with you."

For the support question, you have to analyze the child's total support needs, and how much the child provides for themself.  (You can claim the child if they live with you even if someone else provides support, as long as the hild does not provide more than half their own support).  Support the child needs includes room and board, medical care and insurance, travel, entertainment, and tuition.  Support the child provides themself includes money they spend on these things from their savings or earnings, plus student loans they take out in their name.  (Scholarships don't count as support provided or support needed.). Support provided by others can include the value of room and board you provide them (a share of your rent or mortgage, utilities, food, and so on), tuition you pay, medical insurance if they are on your plan, and so on.

If the child is under age 24, lives at home more than half the year, and does not provide more than half their own support, then you can claim them and they must answer that they can be claimed.