You can also do a slow double click on the layer name in the TOC so that the name is highlighted in a box and enter a new name, similar to renaming a file in Windows Explorer. You can also change the symbology labels under the Label heading on the Symbology tab.
You could change the actual name of the shapefile or feature class, but that will break existing links in the map. You can reset those, of course.
1. Go to layer properties and change Layer name in the General tab.
2. If you want to keep the layer name as is (e.g. to prompt what the data is), go to layer properties and change/add required label to 'Legend' window on Symbology tab. This is the text that appears and replaces what would be labelled otherwise from the layer name.
2. To change labels insitu select layer in the table of contents - this can be the layer name or if you already have label text from 1 above select the label - hit F2 and this should make the text editable. Changing text this way changes what you would see as the layer name of the text entered as per 2 above.
3. You can also add extra labels using the Description - this is accessed from layer properties, symbology tab, click on the Description button. Default behaviour for description is to add the Description text as additional (tacked on after) whatever label is produced from 1 or 2 above, and is added as italics. You can use this to add multiline legend label by copying in multiline text into this description window. Note - editing capabilities in this window are limited so if you want multiline you will need to create the multiline text elsewhere (eg in Notepad) and copy into the descriptions window.
When using Descriptions as a label, to remove the added label from layer name or Legend text (1 and 2 above), go to the style for that layer - right click on the legend, select Properties, select the required layer in the Items tab and click on Style button, Properties button, unclick the 'Show Labels'. At this stage you can change from italics to normal text by clicking on the Description Symbol button.