Change LLC Name
Technically, either option would work fine, since in the end you would be able to do business under the new name. So its mostly a question of cost, convenience and preference. Depending on the state it might be cheaper to do amendment or register a DBA, and the processing times (as well as the actual jurisdiction - state or county), might be different.
From convenience point of view, amending the actual name might require you to update it with other authorities, such as the IRS or the state Department of Revenue (or it's equivalent), as well as updating sales tax ID and licensing information, if your business has any. It's not hard, just something to remember (by the way, its not a bad idea to do the same when registering a DBA, but its not as crucial).
And finally, if you don't mind having your company named the old way while doing business under a DBA then its fine. Sometimes the nature of the business changes so much that you might want to simply amend the name, so that nothing would confuse you, your clients, business partners of any kind, or the authorities.