This is while I'm developing a Lex function but I have had this issue with other services too.
I have a dropdown to select the lambda function but there are no functions available to select in the dropdown
This is frequently due to being in the wrong region.
Lambda may not be available (or that specific function is not created) in the same region as the functionality of the service (Lex in this case).
Solution:
For the lambda function, switch to the same region (N.Virginia in my case), create the function there and then it will appear in the dropdown.