I put GoodYear Eagle F1 All season tires on my 2002 GT convertible. Although these tires are very highly rated, I don't care for the ride.

A convertible does not have the rigid body structure of a coupe and the Kevlar sidewalls on this tire are totally unforgiving of road bumps. This causes a very poor ride and vibrations at highway speeds.
Since you are putting new tires on a convertible , I would caution you to pay particular attention to the sidewall stiffness as you may not care for the resultant ride. I am going to look for another set of tires and sell these. A pretty costly mistake to make!
I got the same tires a few months ago to get through the winter...............very harsh on rough roads, but the tires are awesome in the rain AND got me throguh some snow this past winter. Not sure of heat grip yet but tire rack gave it pretty good ratings.
With tires, you gotta remember that its all about tradeoffs. Get a long life tire? The performance is going to suck. ( ie: lots of Michellin models ) Get an awesome tire for dry heat with great steering response? Not so great in the rain and very average when the temp gets lower ( ie: Conti Sport Contacts ). Get an awesome tire that performance is great in all weather
and solid in most conditions? Real crappy under hard cornering as the tire gets squisy ( Conti-ExtremeContacts - AvonTech AS550 - Summitomo HTR+). G-Force Sports are awesome on dry and in the heat..........very sticky.......but suck at 70 mph on the rain. Ive owned them all.
Personally........I thought that the General Exclaim UHP that happened to come with the car was a pretty damn good tire. The things were definately way sticky for launches imo and pretty good in the rain, wore well and notr real expensive at all. Handling is OK at best on turn-in and hard cornering but not terrible ( Continental ContiSports are awesome if sharp turn in and handling is most important to you )............I think a pretty awesome tire.
Tires Ive had in the past 10 years..............
Continental Conti-Sport Contacts
Conti Extreme Contacts
Bridgestone SO3's
Bridgestone RE750's
Summitomo HTR-+
Avon Tech AS 550
BFG KDW's
Goodyear F1 AS's
General Exclaim UHP
Yokohama Adva S4
All around..........I thought the Generals were probably the best of the bunch given all the tradeoffs...........only wish I had tried them in the snow.
ContiExtreme's are too soft
Summitomo tires wear too fast....................same Avon Tech's/Yokahama
BFG's suck in rain.
SO3's sucked in anything under 50 degree's ( they dont make them anymore anyway )
I gotta do some hard launches with the Goodyears when the temp gets over 85 degrees ( which who the hell knows if that'll ever happen again on Long Island????!!!!) to determine if they are good in dry heat..............but damn they are harsh due to that kevlar stuff.