At a HOA meeting can a non board member make a motion?

Your answer depends on the contents of your governing documents and the state law that governs not only the association, but the form of the corporation that your association fits within, such as the non-profit corporate law.

Generally, the president doesn't vote unless it is to break a tie. However, different associations and different state laws may not agree on this point. Unless specifically forbidden, a president may be able to make a motion, although best practices of good leadership indicate that motions be made by other board members.

Also, generally, board meetings are conducted following Roberts Rules of Order. You can find a quick reference to motions, below.

Finally, motions exhibit decisions for actions. Motions require a percentage of votes, either of the ownership or of the board, regardless of who makes them, or how the president votes.

If you believe that your president has abused her/his power by making a motion and voting on the motion in such a way so as to have violated your governing documents or state law(s), you can formally protest the motion and require further discussion on it. You can also request that the protest be noted in the minutes.

You may fail in your protest, but your protest can be noted in the minutes. Since the minutes are historical documents of the business of the corporation, they become legal evidence.