Create a RadComboBox, bound to a collection of objects, implementing INotifyPropertyChanged. Use DisplayMemberPath. Select an item, then change the display member property of the selected item - the selection box is not updated.
As workaround you can use DataTemplate:
<telerikInput:RadComboBox HorizontalAlignment="Left" Margin="122,52,0,0" VerticalAlignment="Top" MinWidth="150" x:Name="radComboBox" telerik:TextSearch.TextPath="Name">
<telerikInput:RadComboBox.ItemTemplate>
<DataTemplate>
<TextBlock Text="{Binding Name}" />
</DataTemplate>
</telerikInput:RadComboBox.ItemTemplate>
</telerikInput:RadComboBox>