Nearby publics include Costa Mesa CC, which has 36 holes (Los Lagos longer, Mesa Linda shorter and sportier), and the Navy course in Seal Beach, which is one of the courses (I believe "the" primary course) where Tiger learned to play. None of these options are noteworthy from a GCA perspective, however. Same goes for the lone private option in Huntington Beach, The Huntington Club, formerly known as SeaCliff CC. It's decent but nothing worth going out of your way to gain access.
If you head south to Irvine, there are some higher quality/higher dollar options such as Strawberry Farms (Jim Lipe) or Oak Creek (Tom Fazio).
Heading north a bit brings public options like Los Verdes or Terranea (9-hole par 3 resort course by Todd Eckenrode) into play. If you're willing to further enrich the President, his place is around there, too.
How far are you willing to drive? There's really not much noteworthy public golf within 30 minutes or so of HB (outside of Pelican Hill, which I second if you're okay spending $400 or so).
Private-wise, Newport Beach CC is one city south and hosts a PGA Tour Champions event annually, but doesn't look like anything special. (I've never played it.) Newport Beach is also home to Big Canyon, but I believe it's quite difficult to gain access.
Ben's suggestion of Santa Ana CC is probably the best recommendation; I have not played it but it looks fantastic after Jay Blasi's renovation probably ~7 years ago.
Hope this helps.
Edit to add that Long Beach has three pretty good municipal options: Recreation Park, El Dorado Park, and Skylinks. It's also home to the private Virginia Club, another one I've never played but is supposed to be quite good (and is Patrick Cantlay's current or former club, I believe).