Orin Kerr pens Tips
on Inviting Law Professors to Write Amicus Briefs, which is of broader application too. His main points:
1) You're asking for a big favor;
2) Explain why the case matters;
3) Explain how the professor's contribution could be uniquely helpful;
4) Explain if the case is pro bono;
5) Make the initial request by email;
6) Don't hide what the case is about in email attachments;
7) Be polite if the answer you get is 'no'