A third term might have been chosen to maintain the electoral incentive for good performance in a President's 2nd term. As it stands today, two-term Presidents are lame ducks for at least half their second term. They spend much of it "seeking a legacy." If they had the ability to seek another term, perhaps they'd get more done and/or remain more responsive to the public.
The tradition from Washington and the sheer exhaustion of being President and the inevitable pendulum of politics would ensure that few Presidents actually served 3 terms, but the potential of a third term might enliven contests that otherwise appear perfunctory. The huge field of candidates in some primaries might be winnowed if the President continued to control the beginning of the electoral season in his second term.
Similarly, in a President's re-election campaign, he usually runs on a very similar platform with a very similar coaltion behind him. The potenital for longer incumbency might enable a President to change tacks and garner different sources of support. Indeed, the spectre of a third term might invigorate re-election campaigns for a second. Under current law, ambitious candidates sometime wait 4 year for an "open" race. An oppoisition Party has more reason to try to find a good candidate rather than wait for 12 years. The party in power will also have more open primary races because ambitious people in that Party will either challenge the sitting President or force concessions from him.
A good rotation in offices aside, it takes about 18 months for most Presidents to learn their main task (foreign policy), a third term with the ability to change tack allows the retention of talent. Accordingly, perhaps 3 was chosen because that was how long FDR ended up serving (a few months in his fourth before he died) and how long FDR was needed (for the Depression and the for WWII). It also might have been chosen as a better alternative than 2 terms or 10 years: TR couldn't have run for a third term in 1908 nor 1912 under the 22nd Ammendment. While he decided not to 1908 because of his 1904 promise and failed in 1912, he was a good candidate both times. Why limit a successful VP to 10 years, particularly if at times the formula would have him serves less than a "normal" eight.
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All of the above are also reasons to oppose the adoption of the 22nd Ammendment altogether.