Getting my deceased moms property from her storage unit

What do you mean by "she named me and my little brother on the contract"? If you, your brother and your mother were all the signatories to the contract and you are all responsible for paying the storage fee, then they have to let you have access--you have as much right as your mother had because you are parties to the contract. If they don't, you can sue them for a court order that they provide access and/or for the value of the goods they are withholding from you.
But if only your mother signed the contract and was resposible for it, then even if she had indicated that the two of you had the right to access the unit, if only she was a party to it (e.g. only she was responsible to pay), then when she died, your right to access the space ended, too, because you did not have an independent right to access it: you only had a right to access because of her. Once a person passes away, their right to do things ends with them; the contract is over. Therefore, rights derived from them also cease.
That doesn't mean that you don't have a right, as her beneficiaries or heirs, to the unit's contents. But to access it, you need authority from the probate court as the executor (if she had a will naming you executor) or administrator/personal representative (either term may be used) if there was no will. That person, the executor or "PR," has legal right to recover/collect the deceased's belongings. Once you have that authority--contact the probate court to find out how to apply for it--you can show a copy of the court order or "letter testamentary" (as it is often called) giving you legal authority and they should allow you access--if they don't, you could then sue them for access. But be aware that you will have to pay any unpaid storage charges, because they are allowed to hold the contents as security for the charges and sell/auction them if the charges are not paid.