Health Insurance Stability Act of 2008 - Amends the Internal Revenue Code to allow individual taxpayers a refundable tax credit for amounts paid to continue health care coverage for the taxpayer, the taxpayer's spouse, and dependents under the applicable continuation provisions of the Consolidated Omnibus Budget Reconciliation Act (COBRA). Limits the amount of such credit and disqualifies taxpayers who, in any month, have employer-provided coverage, are covered under Medicare, Medicaid, or the State Children's Health Insurance Program (SCHIP) of the Social Security Act, or are imprisoned. Denies such credit to individuals who are not U.S. citizens or permanent residents or who are dependents of a taxpayer. Directs the Secretary of the Treasury to establish a program for advancing payments of credit amounts to health insurance providers of the taxpayer.