Good question! There's no reason that a copy of a lien waiver shouldn't be upheld in a court of law. Nothing in the Florida mechanics lien statute requires that an original copy of a lien waiver be held for some specific amount of time.
Conceivably, the only time that some electronically stored or transmitted copy of a lien waiver might come into question is if the party who submitted the waiver tried to contend that they did not, in fact, submit that waiver. So, having a record of an email or a fax cover sheet that has come along with the waiver might even work to support the fact that the waiver was signed by the party who submitted it.
Futher, even in a situation where a claimant tried to contend that a copy or electronically stored lien waiver wasn't actually submitted by them - they'd essentially be asserting that the party relying on that lien waiver had committed fraud by faking their lien waiver.
With all that being said - that is one benefit of using an electronically signed lien waiver rather than storing emailed, faxed, or even photocopied lien waivers. When using an electronic signature, a record is stored of when the waiver was signed, who signed it, and even the IP address of the location where it was signed. So, it may be easier to prove that the party submitting the waiver was actually the one who signed it.
But, as mentioned above, unless a party who has submitted a waiver does a full heel turn and tries to claim that they weren't the one to sign a waiver, the method of storing or transmitting that waiver - be it via email, fax, photocopy, hard copy, what have you - should be largely irrelevant.