I think the current search engine functionality works well. If you type in just the genus, then the first result will be the genus entry (if it exists). Dave modified it a while ago so that if there is an exact match to the search term, it will be the first result. After that, it is the most popular, which makes sense since most search engines use some kind of algorithm such as popularity to determine relevance of results.
If you want the alphabetical list for a genus, all you need to do is click on the link "View all plants in the genus" at the top.
As for the synonyms that
@sedumzz mentioned, that's part of the Plant Database functionality rather than the search engine. Dave should be able to add a filter option to toggle between the current view and showing only entries whose accepted name is in the selected genus.
