Does this mean that the edifice was a place where people gave testimony, or that the tabernacle itself gave testimony to something?
Ibn Ezra and others explain the name on the basis that this was where the Israelites deposited the “tablets of the testimony” (i.e. the Decalogue, the evidence of the covenant between God and Israel: see Ex. 25:16).
Rashi says that the tabernacle is in itself testimony that God had forgiven the people for making the golden calf.
Had he not pardoned them, He would have held back His presence and not allowed it to dwell either in the people or in the sanctuary.
The Malbim (19th century) adds that the tabernacle was evidence that no-one had breached the requirements of honesty in creating the edifice – neither the Israelites who donated so much to the project, nor Moses in his account-keeping of the donations.