I think "she has been sick this weekend" means that she was sick during the weekend but (probably) isn't sick anymore. I would put that in German like this "Sie war an diesem Wochenende krank." "She became..." or "She got..." (geworden) means that she got sick, and probably still is. So I think the latest are more accurate translations.
I've got bad news, pal: this (ist vs hat) happens a lot. In a nutshell, whenever you want to use the past participle with a verb that indicates a change of state or location, you have to use sein instead of haben.
For example, Ich bin gerannt = I have run Du bist geschwommen = You have swum. Sie ist gefahren = She has driven Ich bin einen Mann geworden = I have become a man.
For us new German learners, it can get really tough, but the key that helps me is to focus on the past participle (gerannt, geschwommen, gefahren, geworden) to realize that it's in the past; not the "haben." Of course, this doesn't help you translate English ---> German using the right helping verb.
I would be interested to read some other takes from other German learners on this rule.
Hmm, this reminds me a little of how the past tense is used in French (passé composé) with verbs indicating movement taking être-to be (eg. Je suis allé) and other verbs taking avoir- to have (eg. Tu as mangé). Does German require use of sein for reflexive verbs in the past tense as well, like in French? (Eg. Je me suis demandé)