Are there any special considerations for extending a UHF-carrying coax wire in a small device?
The coax impedance must be matched and the cable must be able to carry the 1900MHz signal. It might say the impedance on the cable (like 50Ω or 75Ω), if it does then your in luck. If it doesn't then the only way to find out is to test it (which would be very difficult and require specialized equipment).
Additionally the repair would need a continuous wire (so no splicing). Splicing can't match impedance specs. The cable would need to be soldered (or connected) in the same way to the antenna as the cable is now.
The easiest thing to do would be to get a new antenna/cable, but whatever happens the impedance must be matched.