Although Presti's in Little Italy is not a coffeehouse, I think it is a great place to sit and have coffee and a pastry. Nice atmosphere and good food, but if you are looking for fair-trade fresh-ground such and such then this is not your place. Opens early.
Algebra Teahouse (on Murray Hill Rd, also in Little Italy) doesn't open until 10am; it is a way-funky, oh-so-college kind of place. Loose tea, coffee ground on site, living room furniture, patchouli smell, etc.
Phoenix Coffee is up on Coventry, not in University Circle proper, but just a 5 minute drive. They have very good coffee and tea, and always some interesting frou-frou drinks (I think they are serving the red tea honey latte now which is pretty fantastic). More mainstream than Algebra, but still caters to a student crowd. Depending on what time you are there, it could be a sea of laptops.
2136 Murray Hill Rd, Cleveland, OH 44106
1854 Coventry Rd, Cleveland Heights, OH 44118
12101 Mayfield Rd, Cleveland, OH 44106