Google Sites now allows additing of Custom Search Engine as a search option:

1. Login to Google Site, select "Manage Site" option in "More Actions".
2. Select "Site Layout" from Navigation menu on left. Click "Configure Search".
3. Select Advanced tab in "Configure site search". To add a new search provider, click "Add provider".
4. To get information about Custom Search engine to use. Open another tab and login to CSE control panel. Find URL for search engine you want to use by clicking on it's name in the list called My search engines. Copy this URL; you’ll use it in the next step.
5. Go back to Google Sites search provider dialog. Select "Google Custom Search" as provider type. Type in some Name “Some Search”, and paste Custom Search Engine URL obtained in previous step. Leave Show results in site option checked. Click OK.
6. To change default provider click "General" tab in "Configure site search" dialog and selecting "Some Search". Click Save. You're done.

Source:→ Custom Search blog