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I need to make a grid pattern with points that stagger to make a diamond pattern instead of the regular pattern that a fishnet grid makes. It is sometimes referred to a "staggered start". My points need to be 800ft away from each other. I am creating my map in ArcGIS 10.2. Is there something else I can try?

enter image description here

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Is it just a one time job or you need to have it available as a part of a larger script? Would you accept a minor manual after-cleaning? How big is your area (sq km)? – Alex Tereshenkov Mar 15 at 7:23
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Once you get your origins straight, there's no reason you can't create TWO fishnets and append the results. Or just write a short arcpy script to create the points directly. – Vince Mar 15 at 10:50
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I still can use the free Hawth's Tools in ArcGIS with the VBA extension although that isn't recommended. It has a tool to do what you want. The author created a newer set of tools for use in ArcGIS 10x and here's a link to that tool: spatialecology.com/htools/regpnts.php but I haven't tested it. I use ET Geowizards to create my point grids. I have an older paid version but I think this is one of its free tools. – johns Mar 15 at 17:38

As far as I know there isn't a specific tool to do that, but others have already built tools for the creation of isometric grids.

I found this toolbox you can download and use. I tested it a few times and it seems to work well in general, but you might still need to perform some manual operations of clipping the points layers, or generating new centroids from the resulting hexagonal grid. enter image description here

Edit: if you need 800 ft between each two points, set the hexagon side distance to 461.8802154 ft. (basic euclidean geometry). Scale up or down depending on your CRS

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It looks like this solution won't allow changing how much every other row is "staggered." Since it only creates points inside regular hexagons, the angle between the first point on one row and the first point on the next row can't be changed. As long as 60 degrees from point to point is what you want, this tool is great, otherwise I like @Vince's solution in the OP comments (which should be made into an answer). – LeonB Mar 15 at 13:31

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