All formal requirements for federations that want to become eduGAIN members and thus add Identity Providers and Service Provider to eduGAIN are described in the eduGAIN Constitution, Section 3 "eduGAIN Membership". In particular section 3.1 "Requirements for Participant Federations" lists the relevant points to consider.
The relevant extract is (please consult the eduGAIN Constitution for the current an up-to-date version):
Participant Federations MUST:
If these requirements are met, a federation can initiate the "Joining Process" described in section 3.2. Some more details on this process are described on described on the page Process for an Identity Federation to Join eduGAIN.