アニメーション付きアンダーラインはAnimatedContainerで表現する。
Text Widgetのwidth取得方法はこちらを参考に実装。

header_menu_item.dart

class HeaderMenuItem extends StatefulWidget {
  final String text;
  final double fontSize = 16;

  const HeaderMenuItem({Key key, this.text}) : super(key: key);

  @override
  _HeaderMenuItemState createState() => _HeaderMenuItemState();
}

class _HeaderMenuItemState extends State<HeaderMenuItem> {
  double _opacity = 0;
  double _lineRatio = 0.5;

  @override
  Widget build(BuildContext context) {
    return MouseRegion(
      onEnter: (details) => _onEnterHandler(details, context),
      onExit: (details) => _onExitHandler(details, context),
      child: Padding(
        padding: const EdgeInsets.symmetric(horizontal: 30.0),
        child: Column(
          crossAxisAlignment: CrossAxisAlignment.center,
          children: [
            Expanded(
              child: Container(
                alignment: Alignment.center,
                child: Text(
                  widget.text,
                  style: TextStyle(
                    fontSize: widget.fontSize,
                    letterSpacing: 5,
                  ),
                ),
              ),
            ),
            AnimatedContainer(
              width: lineWidth(widget.text) * _lineRatio,
              height: 2,
              color: Colors.black.withOpacity(_opacity),
              curve: Curves.linear,
              duration: Duration(milliseconds: 200),
            ),
          ],
        ),
      ),
    );
  }

  // マウスホバー時
  void _onEnterHandler(PointerEvent details, BuildContext context) {
    setState(() {
      _lineRatio = 1;
      _opacity = 1;
    });
  }

  // マウスホバー解除時
  void _onExitHandler(PointerEvent details, BuildContext context) {
    setState(() {
      _lineRatio = 0.5;
      _opacity = 0;
    });
  }

  // TextWidgetのwidthを取得
  double lineWidth(String text) {
    var renderParagraph = RenderParagraph(
      TextSpan(
        text: text,
        style: TextStyle(
          fontSize: widget.fontSize,
          letterSpacing: 5,
        ),
      ),
      textDirection: TextDirection.ltr,
    )..layout(BoxConstraints(maxWidth: 200));

    return renderParagraph.getMinIntrinsicWidth(widget.fontSize);
  }
}

参考