This plugin offers an assortment of flags that can replace that of the logged-in site. Each choice is saved immediately. Choose often. Refresh offers new choices.
Tip: complete a welcome page with two links, one to the doing that is easy to share and the other to the author's statement of purpose and qualification. The statement usefully includes the rationale for the chosen site name and memory mnemonics for the flag.
Plugin source can be found on github 
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The item contains a count of "choices" but offers no way to adjust it. 40 seems a good number.