From Evernote:

AndroidのWebViewでLocalStorageを使う為の設定

String databasePath = this.getApplicationContext().getDir("localstorage", Context.MODE_PRIVATE).getPath();

Log.d(LOG_TAG, "databasePath=" + databasePath);

webView = (WebView) findViewById(R.id.webview);

webView.getSettings().setJavaScriptEnabled(true);

webView.getSettings().setDomStorageEnabled(true);

webView.getSettings().setDatabasePath(databasePath);

// webView.setWebChromeClient(new BabukumaWebClient());

webView.loadUrl(URL_TOP);