Is it weird to say either 我今天休息了一天 or 我今天休息一天了 ("today, I rest for one day") because of the repetition of 天?
No. It is not weird at all.

我今天休息 means: I rest/will rest today. It usually is only saying that: I do/will not work today.

我今天休息了 means: I have taken a rest today. It shows no information about how long I rested, and usually not a whole day. Usually, it is used to say that I have already taken a rest during my work/study/exercise/... today.

Both the given examples are saying that I've rested for a whole day.