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For small groups you can use the following naive approach:

E1=EllipticCurve(GF(8209),[1,0,0,333,6166])
ord=E1.abelian_group().order()
o=0
while o<ord: P=E1.random_element();o=P.order();
print "P: ",P
print "order: ", o
#P:  (2388 : 3065 : 1)
#order:  8210

E1=EllipticCurve(GF(8209),[1,0,0,333,6166])
ord=E1.abelian_group().order()
o=0
while o<ord: P=E1.random_element();o=P.order();
print "P: ",P
print "order: ", o
#P: E1=EllipticCurve(GF(8209),[1,0,0,333,6166])  (2388  
P=E1.gens();P
#[(7101 : 3065 5224 : 1)
#order:  8210
1)]
P[0].order()
#8210

E1=EllipticCurve(GF(8209),[1,0,0,333,6166])   
P=E1.gens();P
#[(7101 : 5224 : 1)]
P[0].order()
#8210
sage: E1=EllipticCurve(GF(8209),[1,0,0,333,6166])
sage: P=E1.gen(0)
sage: P.order()
8210

sage: E1=EllipticCurve(GF(8209),[1,0,0,333,6166])
sage: P=E1.gen(0)
P=E1.gen(0);P
(3714 : 7019 : 1)
sage: P.order()
8210