Does anybody know which practical use cases there are to operate on encrypted strings? Even niche problems that can be solved using homomorphic encryption on strings are interesting to me, but not academic ones.
I believe you're talking about homomorphic encryption. It allows to have operations on encrypted data.
A simple use-case example:
Sometimes data has to be kept confidential (i.e. names, address, ... ). It could be that you want to perform computing on data but you're missing the computational power to do so. So you're outsourcing this computation to someone else (cloud computing), but now you have a problem with privacy and confidentiality, because in this case the data has to be handled confidentional.
That's where homomorphic encryption is useful. You encrypt the data and afterwards an untrusted server can perform computation on this data. After the computation you can decrypt the data again and a third party has not gained any knowledge of the data.
Examples in the case of strings:
You have a database with tables and you would like to add a new row with data. The database can be encrypted and hosted by an untrusted server (we're looking over the fact that this is maybe a bad practice). You can encrypt the data and add it to the database.
You want to compare a lot of data (strings), for example you want to be sure that database A ($D_a$) has exactly the same data as the database B ($D_b$). You can compare the data with homomorphic encryption by an untrusted third party. After the comparison you know if $D_a = D_b$.
I'm surprised that nobody mentioned genomics as use case. In this field, the underlying strings to manipulate are DNA strings like "AACTGA...". Homomorphic encryption can be useful to manipulate encrypted DNA strings on an untrusted cloud, while still maintaining data confidentiality.
For more info, see here for typical string operations.