Custom Adapter with Filter

Chemical.cs

namespace SearchViewSample
{
    public class Chemical
    {
        public string Name { get; set; }

        public int DrawableId { get; set; }
    }
}

ChemicalsAdapter.cs

using System.Collections.Generic;
using System.Linq;
using Android.App;
using Android.Views;
using Android.Widget;
using Java.Lang;
using Object = Java.Lang.Object;

namespace SearchViewSample
{
    public class ChemicalsAdapter : BaseAdapter<Chemical>, IFilterable
    {
        private List<Chemical> _originalData;
        private List<Chemical> _items; 
        private readonly Activity _context;

        public ChemicalsAdapter(Activity activity, IEnumerable<Chemical> chemicals)
        {
            _items = chemicals.OrderBy(s => s.Name).ToList();
            _context = activity;

            Filter = new ChemicalFilter(this);
        }

        public override long GetItemId(int position)
        {
            return position;
        }

        public override View GetView(int position, View convertView, ViewGroup parent)
        {
            var view = convertView ?? _context.LayoutInflater.Inflate(Resource.Layout.Chemical, null);

            var chemical = _items[position];

            var nameView = view.FindViewById<TextView>(Resource.Id.chemName);
            var imageView = view.FindViewById<ImageView>(Resource.Id.chemImage);
            nameView.Text = chemical.Name;
            imageView.SetImageResource(chemical.DrawableId);

            return view;
        }

        public override int Count
        {
            get { return _items.Count; }
        }

        public override Chemical this[int position]
        {
            get { return _items[position]; }
        }

        public Filter Filter { get; private set; }

        public override void NotifyDataSetChanged()
        {
            // If you are using cool stuff like sections
            // remember to update the indices here!
            base.NotifyDataSetChanged();
        }

        private class ChemicalFilter : Filter
        {
            private readonly ChemicalsAdapter _adapter;
            public ChemicalFilter(ChemicalsAdapter adapter)
            {
                _adapter = adapter;
            }

            protected override FilterResults PerformFiltering(ICharSequence constraint)
            {
                var returnObj = new FilterResults();
                var results = new List<Chemical>();
                if (_adapter._originalData == null)
                    _adapter._originalData = _adapter._items;
                
                if (constraint == null) return returnObj;

                if (_adapter._originalData != null && _adapter._originalData.Any())
                {
                    // Compare constraint to all names lowercased. 
                    // It they are contained they are added to results.
                    results.AddRange(
                        _adapter._originalData.Where(
                            chemical => chemical.Name.ToLower().Contains(constraint.ToString())));
                }
                
                // Nasty piece of .NET to Java wrapping, be careful with this!
                returnObj.Values = FromArray(results.Select(r => r.ToJavaObject()).ToArray());
                returnObj.Count = results.Count;

                constraint.Dispose();

                return returnObj;
            }

            protected override void PublishResults(ICharSequence constraint, FilterResults results)
            {
                using (var values = results.Values)
                    _adapter._items = values.ToArray<Object>()
                        .Select(r => r.ToNetObject<Chemical>()).ToList();
                    
                _adapter.NotifyDataSetChanged();

                // Don't do this and see GREF counts rising
                constraint.Dispose();
                results.Dispose();
            }
        }
    }
}

SearchViewActivity.cs

using System.Collections.Generic;
using Android.App;
using Android.Runtime;
using Android.Support.V4.View;
using Android.Support.V7.App;
using Android.Support.V7.Widget;
using Android.Views;
using Android.Widget;
using Android.OS;

namespace SearchViewSample
{
    [Activity(Label = "SearchView Sample", MainLauncher = true, Icon = "@drawable/icon",
        Theme = "@style/Theme.AppCompat.Light")]
    public class SearchViewActivity : ActionBarActivity
    {
        private SearchView _searchView;
        private ListView _listView;
        private ChemicalsAdapter _adapter;

        protected override void OnCreate(Bundle bundle)
        {
            base.OnCreate(bundle);

            SetContentView(Resource.Layout.Main);

            SupportActionBar.SetDisplayShowHomeEnabled(true);

            var chemicals = new List<Chemical>
            {
                new Chemical {Name = "Niacin", DrawableId = Resource.Drawable.Icon},
                new Chemical {Name = "Biotin", DrawableId = Resource.Drawable.Icon},
                new Chemical {Name = "Chromichlorid", DrawableId = Resource.Drawable.Icon},
                new Chemical {Name = "Natriumselenit", DrawableId = Resource.Drawable.Icon},
                new Chemical {Name = "Manganosulfate", DrawableId = Resource.Drawable.Icon},
                new Chemical {Name = "Natriummolybdate", DrawableId = Resource.Drawable.Icon},
                new Chemical {Name = "Ergocalciferol", DrawableId = Resource.Drawable.Icon},
                new Chemical {Name = "Cyanocobalamin", DrawableId = Resource.Drawable.Icon},
            };

            _listView = FindViewById<ListView>(Resource.Id.listView);
            _adapter = new ChemicalsAdapter(this, chemicals);
            _listView.Adapter = _adapter;
        }

        public override bool OnCreateOptionsMenu(IMenu menu)
        {
            MenuInflater.Inflate(Resource.Menu.main, menu);

            var item = menu.FindItem(Resource.Id.action_search);

            var searchView = MenuItemCompat.GetActionView(item);
            _searchView = searchView.JavaCast<SearchView>();

            _searchView.QueryTextChange += (s, e) => _adapter.Filter.InvokeFilter(e.NewText);

            _searchView.QueryTextSubmit += (s, e) =>
            {
                // Handle enter/search button on keyboard here
                Toast.MakeText(this, "Searched for: " + e.Query, ToastLength.Short).Show();
                e.Handled = true;
            };

            MenuItemCompat.SetOnActionExpandListener(item, new SearchViewExpandListener(_adapter));

            return true;
        }

        private class SearchViewExpandListener 
            : Java.Lang.Object, MenuItemCompat.IOnActionExpandListener
        {
            private readonly IFilterable _adapter;

            public SearchViewExpandListener(IFilterable adapter)
            {
                _adapter = adapter;
            }

            public bool OnMenuItemActionCollapse(IMenuItem item)
            {
                _adapter.Filter.InvokeFilter("");
                return true;
            }

            public bool OnMenuItemActionExpand(IMenuItem item)
            {
                return true;
            }
        }
    }
}

If people don't want to use the "nasty object wrapping", you can avoid it if you let your class extend Java.Lang.Object itself like:
public class Chemical : Java.Lang.Object

You'd then change the relevant parts in your FilterResults and PublishResults methods:
returnObj.Values = results.ToArray();

_adapter._items = new List<Chemical>(values.ToArray<Chemical>());