Anyone? The question is:

Is it possible to search part of filename using the file manager search?

You could implement a custom search provider, like this:

    [Export(typeof(ISearchProvider))]
    public class CustomFileSearchProvider : ISearchProvider
    {
        private static ILog _log = LogManager.GetLogger(typeof(CustomFileSearchProvider));

        public string Area
        {
            get
            {
                return "CMS";
            }
        }

        public string Category
        {
            get
            {
                return "Filenames";
            }
        }
        public IEnumerable<SearchResult> Search(Query query)
        {
            var list = new List<SearchResult>();
            foreach (string virtualDir in (IEnumerable<string>)VirtualPathHandler.Instance.VirtualPathToNameMapping.Keys)
            {
                try
                {
                    UnifiedDirectory unifiedDirectory = HostingEnvironment.VirtualPathProvider.GetDirectory(virtualDir) as UnifiedDirectory;
                    if (unifiedDirectory != null)
                    {
                        var filez = unifiedDirectory.Files;
                        var files = new List<UnifiedFile>();
                        GetAllFiles(unifiedDirectory, files);
                        list.AddRange(from file in files where file.Name.Contains(query.SearchQuery) select CreateSearchResult(file.VirtualPath, file.Name));
                    }
                }
                catch (Exception ex)
                {
                    _log.Warn((object)"There was an error when trying to perform a search for filenames.", ex);
                }
            }
            return list;
        }

        private void GetAllFiles(UnifiedDirectory start, List<UnifiedFile> files)
        {
            foreach (UnifiedFile file in start.GetFiles())
            {
                files.Add(file);
            }
            foreach (UnifiedDirectory dir in start.GetDirectories())
            {
                GetAllFiles(dir, files);
            }
        }

        private SearchResult CreateSearchResult(string path, string name)
        {
            return new SearchResult(UriSupport.ResolveUrlFromUIBySettings(string.Format("edit/default.aspx?plugin={0}&selectedfile={1}", (object)PlugInDescriptor.Load("EPiServer.UI.Hosting.VirtualPathControl", "EPiServer.UI").ID, (object)HttpUtility.UrlEncode(path))), name, null)
            {
                IconCssClass = this.GetIcon(name)
            };
        }

        private string GetIcon(string fileName)
        {
            return "epi-resourceIcon epi-resourceIcon-" + Path.GetExtension(fileName).TrimStart(new char[1]
      {
        '.'
      });
        }
    }

Thanks a bunch! I'll try that!

How can I configure Episerver to use my custom files search provider? Could you point me to any documentation on this?

Mark your class with a [SearchProvider] attribute, and make sure you have added a reference to System.Component.Composition.

More documentation can be found here:

http://world.episerver.com/Blogs/Magnus-Stalberg/Dates/2009/10/Implementing-a-Search-provider-for-SiteCenter-in-5-minutes/
http://world.episerver.com/Documentation/Items/Tech-Notes/EPiServer-CMS-6/EPiServer-CMS-60/OnlineCenter-Developer-Documentation/#AddingSearchProviders