There is much to be understood, before explaining what the testimony actually is; Since the Hebrews had been
enslaved in Egypt they had forgotten how to serve God, and God; through Moses; was teaching them how he was to be worshipped.
We must begin in the book of Exodus and build up to answering your question.
All Scripture is quoted from the King James version.
Exodus 25:10 And they shall make an ark of shittim wood: two cubits and a half shall be the length thereof, and a cubit and a half the breadth thereof, and a cubit and a half the height thereof.
Exodus 25:16 And thou shalt put into the ark the testimony which I shall give thee.
Notice that it says the testimony which I will give thee
According to Strong's dictionary, the translation here is from the words:
עדוּת‛ êdûth ay-dooth'
Feminine of H5707; testimony: - testimony, witness.
עד ‛êd ayd
concretely a witness; abstractly testimony; specifically a recorder, that is, prince: - witness.
Exodus 25:17 through 22 And thou shalt make a mercy seat of pure gold: two cubits and a half shall be the length thereof, and a cubit and a half the breadth thereof. 18 And thou shalt make two cherubims of gold, of beaten work shalt thou make them, in the two ends of the mercy seat. 19 And make one cherub on the one end, and the other cherub on the other end: even of the mercy seat shall ye make the cherubims on the two ends thereof. 20 And the cherubims shall stretch forth their wings on high, covering the mercy seat with their wings, and their faces shall look one to another; toward the mercy seat shall the faces of the cherubims be.
21 And thou shalt put the mercy seat above upon the ark; and in the ark thou shalt put the testimony that I shall give thee. 22 And there I will meet with thee, and I will commune with thee from above the mercy seat, from between the two cherubims which are upon the ark of the testimony, of all things which I will give thee in commandment unto the children of Israel.
The Mercy seat was where God met with Moses to give his explanations of exactly what the Commandments entailed.
Exodus 24:12 And the LORD said unto Moses, Come up to me into the mount, and be there: and I will give thee tables of stone, and a law, and commandments which I have written; that thou mayest teach them.
These precepts of God which he wanted Moses to teach to the people, were summarized on the stone tablets and written by the hand of God himself, and that was repeated by God on the second set of stones.
Exodus 34:1 And the LORD said unto Moses, Hew thee two tables of stone like unto the first: and I will write upon these tables the words that were in the first tables, which thou brakest.
God commanded that Moses put those stone tablets into the Ark, and thereby comes the name Ark of the testimony. Which Moses did in:
Exodus 40:20 And he took and put the testimony into the ark, and set the staves on the ark, and put the mercy seat above upon the ark:
In your question you misstated by asking where Aaron's rod was stored. His rod was not stored in the testimony, but rather with the testimony in the Ark of the Covenant.