"Does the ordinary income in Box 3 of the Form 1099-MISC count as earned income"

No. Box 3 specifically is Other Income. Not earned. Earned income would be in box 7, for being in business income as Non-employee Compensation.

"She only has $2,400 of W-2 income and made a $7,000 Roth contribution. "

That's a problem; there still is time to notify the administrator to send the overage back. There might be a bit of earnings that are taxable, or maybe not, given the current condition of the market.

"she made an Excess Roth IRA contribution since it is not including the ordinary income from exercising the stock options."

The program is right. You are confusing Taxable Income and Earned Income.