How can I create a DOI for a paper that was uploaded to arXiv but not published somewhere else?
Ultimately, no, you can't. There's no infrastructure to say "I want to get a single DOI for XYZ arbitrary url".

DOIs for most scholarly publications are issued through CrossRef. CrossRef do not assign DOIs directly, but delegate this to members or their agents. Members, who are usually publishers, pay a fee to the central consortium, and agree to issue DOIs based on a fixed set of rules.

As you can see from these rules, the general sense of membership is "organisations issuing DOIs for stuff they control". While there isn't an explicit prohibition against assigning DOIs to third-party material (which surprised me!), point 4 comes pretty close:


  
  Members have an obligation to maintain the metadata and URLs associated with all registered DOIs.
  


There is also a general prohibition of 'duplicative' works including preprints:


  
  Crossref only registers DOIs for Definitive Works (or Versions of Record, if not formally published) but not for Duplicative Works, as defined in the Crossref Glossary. This means that only original scholarly material, for which there is no actual DOI at the time of submission, and no expected duplication in future, is admissible for Crossref DOI registration. Crossref does not permit multiple DOIs to be assigned to certain closely related versions of a work, and hence does not support assignment of DOIs to Pre-prints or Post-prints of Definitive Works or to the Personal Version or a Self-archived Copy of a Definitive Work. For the same reasons, materials for which DOI duplication can be reasonably anticipated, such as an Authors Original Draft of a work being prepared for publication, are not admissible for Crossref DOI registration.
  


Putting those together, it seems likely that CrossRef's terms would prevent a member agreeing to issue a DOI for a (potentially duplicative) work hosted somewhere out of their control.

So, could you get a DOI from someone else, outside of CrossRef? Probably not. DOIs are only issued through a number of central registration authorities (eg CrossRef, DataCite). These have fairly well-defined areas of activity (eg DataCite won't issue DOIs for publications) and, to the best of my knowledge, none offer a "DOI for an arbitrary URL" service.