"Extradition" simply means that the Florida county prosecutor's office will make a decision as to whether or not they wish to expend the money necessary to bring your family member back to Florida after he completes his sentence in Ohio to face prosecution for whatever crime he is being charged with in Florida (whatever the outstanding warrant is for in FL).
The time frame for them to make this decision is between 30 and 60 days of your family member being released - if he has more than this amount of time remaining on his sentence, the prosecutor's office has no obligation to rush their decision.
If your family member is being charged with a serious felony, I would highly recommend retaining a private criminal defense attorney that has some familiarity with extradition proceedings (a FL attorney) - this is not something that all criminal defense attorneys handle, but most will be able to. It is possible to "beat" an extradition hearing on the procedure (get around the 30 or 60 day timeline if the FL DA is not careful) - this does not mean the warrant goes away, but your family member may be able to avoid extradition at least for a little while. (Again, if he is charged with a serious felony, this may not be a realistic option - but again, having a good defense attorney early on is his best option if he is facing significant charges).
If he is facing minor charges, the FL DA may decide that the cost of doing an interstate extradition is not worth it, and will not pursue the extradition at this time - again, the warrant does not go away, but your family member may avoid extradition for awhile (unfortunately, there is no way I can tell you whether or not your family member's situation would be one that the DA would ignore or pursue).