If one has had some tendenitis from levaquin, (levofloxacin) should it be prescribed again?

No. Fluoquinolones usually cause tendon ruptures rather than tendinitis. The exact mechanism whereby this happens is poorly understood. Perhaps, it is caused by tendinitis, although, this is a matter of conjecture. One way or the other, if Levaquin (levofloxacin) was suspected as a culprit in the tendon problem, i'd abstain from it next time, unless no other alternative exists.
Quinolone tendonitis. Tendonitis has been reported with all quinolone drugs which include Cipro (ciprofloxacin) and Levaquin and others. It is more common in patients very physically active, prior tendon injuries or surgery. It can result in crippling ruptures including the Achilles tendon. The tendonitis can last months after the drug is stopped. It is not recommended to take Levaquin again with a prior history of tendonitis.
Talk to prescriber. You should always voice your concerns to your prescribing doctor about a medication that you have previously had a reaction or side effect to. There may be an alternative, or perhaps this reaction was not in your record. You are your own best advocate.