How can I alter an existing Primary Key on SQL Azure?
Note: as of Azure SQL Database v12, these restrictions no longer apply.

The is no such thing as a 'primary index'. There is such a thing as a 'primary key' and also there is such a thing as a 'clustered index'. Distinct concepts, often confused. With this distinction in mind, lets revisit the question:

Q1) Can the clustered index in a SQL Azure table be modified?
A: Yes. Use  WITH (DROP_EXISTING=ON):

create table Friend (
    UserId int not null,
    Id int not null);
go  
create clustered index cdxFriend on Friend (UserId, Id);
go
create clustered index cdxFriend on Friend (Id, UserId) with (drop_existing=on);
go


Q2) Can the clustered index of a table that has a primary key constraint be modified?
A: Yes, same as above, as long as the primary key constraint is not enforced via the clustered index:

create table Friend (
    UserId int not null,
    Id int not null identity(1,1),
    constraint pk_Friend primary key nonclustered (Id));
create clustered index cdxFriend on Friend (UserId, Id);
go
create clustered index cdxFriend on Friend (Id, UserId) with (drop_existing=on);
go


Q3) Can the primary key constraint of a table be modified?
A: Yes, as long as the primary constraint is not enforced via the clustered index:

create table Friend (
    UserId int not null,
    Id int not null identity(1,1),
    constraint pk_Friend primary key nonclustered (Id));
go
create clustered index cdxFriend on Friend (UserId, Id);
go
alter table Friend drop constraint pk_Friend;
alter table Friend add constraint pk_Friend primary key nonclustered (UserId)
go


Q4) Can the primary key of a table be modified when is enforced via the clustered index?
A: Yes, if the table never had any rows:

create table Friend (
    UserId int not null,
    Id int not null identity(1,1),
    constraint pk_Friend primary key clustered (UserId, Id));
go
alter table Friend drop constraint pk_Friend;
alter table Friend add constraint pk_Friend primary key clustered (Id, UserId)
go


Q5) Can the primary key of a table be modified when is enforced via the clustered index if the table is populated?
A: No. Any operation that converts a populated clustered index into a heap will be blocked in SQL Azure, even if the table is empty:

create table Friend (
    UserId int not null,
    Id int not null identity(1,1),
    constraint pk_Friend primary key clustered (UserId, Id));
go
insert into Friend (UserId) values (1);
delete from Friend;
go
alter table Friend drop constraint pk_Friend;


As a side note: the constraint can be modified if the table is truncated. 

The workaround to change the PK constraint of a populated table is to do the good old sp_rename trick:

create table Friend (
    UserId int not null,
    Id int not null identity(1,1),
    constraint pk_Friend primary key clustered (UserId, Id));
go
insert into Friend (UserId) values (1);
go

create table FriendNew (
    UserId int not null,
    Id int not null identity(1,1),
    constraint pk_Friend_New primary key clustered (Id, UserId));
go

set identity_insert FriendNew on;
insert into FriendNew (UserId, Id) 
select UserId, Id
from Friend;
set identity_insert FriendNew off;
go

begin transaction
exec sp_rename 'Friend', 'FriendOld';
exec sp_rename 'FriendNew', 'Friend';
commit;
go

sp_help 'Friend';


The sp_rename approach has some issues, most importantly being that permissions on the table do not carry over during the rename, as well as foreign key constraints.