Another example of a good early re-make is THE FRONT PAGE. Originally made in 1931 with Adolf Menjou and Mary Brian, this one has been re-made three times and each version has been excellent. My favorite is HIS GIRL FRIDAY (1940) with Cary Grant and Rosalind Russell. 1974's THE FRONT PAGE and 1988's SWITCHING CHANNELS were good also.
I DIED A THOUSAND TIMES (1955) and COLORADO TERRITORY (1949) were re-makes of the Humphrey Bogart hit HIGH SIERRA (1941). Although the 1955 version is usually panned, I like Jack Palance in the Roy "Mad Dog" Earle part and the 1949 version is good with Joel McCrea. Other decent remakes are PYGMALION (1937) into MY FAIR LADY (1964), and CAPE FEAR (1962, 1991).