angularjs : filter an array with a combobox

The problem is that once you select the empty option, the modelValue becomes null.
Thus, passing the object {date: null} to filter, it will try to much items with date: null.

To satisfy this specific requirement of yours (i.e. treat a null value in search as if it was not defined at all), you could define a function that receives an item and (based on the search object) determines if it should be filtered out or not.

Then you can use that predicate function as an argument to Angular's filter filter.

E.g.:

<tr ng-repeat="budget in filteredBudgets | filter:filterBySearch">

$scope.filterBySearch = function (item) {
    return Object.keys($scope.search || {}).every(function (key) {
        var value = $scope.search[key]; 
        return (value === undefined) || 
               (value === null) ||
               value === item[key];
    });
};

See, also, this short demo.

A slightly more complex demo is available here. It allows a strict parameter to be passed to the predicate function.

UPDATE

An alternative approach would be to implement a custom directive (and add it to the <select> element) with the following functionality:

It will watch for changes in the model-value and convert any null values to undefined.
This solves the problem, because if a property of search has a value of undefined it is as if it is not there at all.

E.g.:

<select ... null-is-undefined>...</select>

app.directive('nullIsUndefined', function () {
    return {
        restrict: 'A',
        require: 'ngModel',   // <-- this is required for accessing the ngModelController
        link: function postLink(scope, elem, attrs, modelCtrl) {
            modelCtrl.$parsers.push(function (newViewValue) {
                if (newViewValue === null) {
                    newViewValue = undefined;
                }
                return newViewValue;
            });
        }
    };
});

See, also, this short demo.