How can I use a simple Dropdown list in the search box of GridView::widget, Yii2?

You can also use below code

[
    'attribute'=>'attribute name',
    'filter'=>array("ID1"=>"Name1","ID2"=>"Name2"),
],

OR

[
    'attribute'=>'attribute name',
    'filter'=>ArrayHelper::map(Model::find()->asArray()->all(), 'ID', 'Name'),
],

Add this in Gridview columns array:

[
    'attribute' => 'attribute_name',
    'value' => 'attribute_value',
    'filter' => Html::activeDropDownList($searchModel, 'attribute_name', ArrayHelper::map(ModelName::find()->asArray()->all(), 'ID', 'Name'),['class'=>'form-control','prompt' => 'Select Category']),
],

Change values according to your attributes.