First of all your problem does not sound expensive. A tire doing that is common but usually will keep the noise up even when holding a fixed speed while still cold. Others experiences will vary of course. You mentioned only when accelerating so I thought it elimiated that. Since you have mentioned it seems to be in the front area it is now a good possibility. There is just not that many things up there that will do this. So move the front wheel to the rear or put the spare on one front side and see if it is gone. If not move the spare to the other side and see. If it is not one front wheel I would still check the rubber disk at the front of the driveshaft carefully for integrity as well as the centre driveshaft mount. May as well have a look at the transmission rubber mount as well to make sure it is not totally oil soaked and coming apart or just rotten rubber. This car is getting older and these parts will die from old age alone sometimes. A fairly common noise generator is if the bracket that fastens the exhaust pipe to the bellhousing area is loose it might kind of rattle more when colder and accelerating. Just check that by pulling on the pipe to make sure it is tight there.