If you're using the New Responsive Designer, there is a "Display unique values only" option in the List Lookup control settings that will allow you to display only unique values in the dropdown list.

I'm using 2013. Looking but I can't find this feature. 

Sadly there is no real easy way of doing what it is you're trying to do. It's easier in fact to reorganize the data so that the list you're looking up from is designed so that each item is unique. That is, create a list called Specialties where each item (Specialty) is unique. 

This is especially true if you're using the result of whatever is selected in that Specialty Dropdown as a value-to-filter-on in a different Lookup Control. 

The reason for this involves the backend on how one filters a lookup query. For instance, if you have another Lookup being filtered by whatever you select from the Specialty dropdown, because the filters are relying on the underlying ID value of the selection in question and not the human readable text, each item then HAS to be displayed because each one of those entries resolves to a different ID.

If you're NOT using the Lookup Dropdown to do any filtering with of any kind, and just want a list of options, create a Choice Control. That comes with the added baggage of needing to update in the event of a new Specialty being created, but if those options are pretty set in stone with no risk of moving around, then it could also solve the problem. 

If it's the case that you want the org to update the options, then yes, I would create a new list that you can lookup from with just the unique values. 

Note that if you do this, you will possibly need to restructure other things if these values are being used in other lookups / lookup columns for cascading lookup purposes. 

In other words, if you have a list that is being filtered by the Specialty value - say a list of Specialists that has a lookup column containing each Specialty they may have, creating a new Specialties List with unique values will not work as a filter unless you retool the Specialists List to use the Lookup Values from your newly created Specialties List (with unique Specialty values). 

Hope that makes enough sense, but I figured it might be worth saying since it's not entirely obvious that it needs to be done. 

 Makes sense to my novice mind and attempting it now. Thanks again