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Do we have any sources addressing which of Moshe Rabbeinu's older siblings, Miriam and Aaron, was older?

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Tosafot Bechorot 4a:

דלא היה אהרן בכור דמרים גדולה ממנו ג' שנים כדאמר במדרש [ריש פ' שמות] כמו שהיה אהרן גדול ממשה ג' שנים

Aharon was not the firstborn, because Miriam was three years older than him, as the Midrash states, just like Aharon was three years older than Mosheh.

The Midrash that Tosafot refers to seems to be Shemot Rabbah 1:13:

וְלֹא הָיוּ לְמִרְיָם אֶלָּא חָמֵשׁ שָׁנִים, שֶׁאַהֲרֹן גָּדוֹל מִמּשֶׁה שָׁלשׁ שָׁנִים

Miriam was only five years old [at the time of Pharaoh's decree against the Hebrew boys] as Aharon was three years older than Mosheh.

Conversely, Ramban to Shemot 15:20:

ואמר אחות אהרן דרך כבוד לו, שהוא אחיה הגדול ואחותו הנביאה מתיחסת אליו

Referring to Miriam as Aharon's sister is a way of honoring him because he is her elder brother, and his sister the prophetess is to be identified through him.

However, as noted by Alex in a comment, it's possible that Ramban does not mean that Aharon was older than Miriam, but just that he was the elder of her two brothers.

See also Rashbam to Shemot 15:20:

אחות אהרן - על שם הבכור קורא אחות

'[Miriam] the sister of Aharon' - the sister is known by the name of the firstborn

which implies that Aharon was older than Miriam.

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    I have seen a reference that Miriam was 13 years older than Moshe Rabbeinu. This would be because she is identified a Puah, which assumes that she was bas mitzvah at that time and the incident with Par'o and the meyaldos occured a year before Moshe was born. I cannot put it as a separate answer because it is from memory. The statement of a year was because it was connected with the reward of houses that they were given. I also saw a reference that says that Miriam was 93 at the Exodus (again from memory).
    – sabbahillel
    Sep 17, 2018 at 14:23
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    @Alex Ramban is cited at mg.alhatorah.org/Full/Shemot/15.20#e0n6 as saying ויתכן שדרך הכתובים ליחס אל גדול האחים, וכן: ובני כלב אחי ירחמאל (דברי הימים א ב׳:מ״ב), שהיה גדול מאחיו, which I read as saying that the reference is to the eldest brother, not the brother who is older than her.
    – sabbahillel
    Sep 17, 2018 at 14:34
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    @JoelK Amazing! I wondered why this question was so hard...
    – SAH
    Sep 17, 2018 at 16:55
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    It wouldn't makes sense that Miriam is younger than Aharon. She had to be the oldest of the three. During the gezerah of Pharoah, Miriam convinced her father to go back to her mother. This produced Moshe about 6 months later. That would make Aharon (maximally) 2 and 1/2 years old at the time Miriam spoke. If so, Miriam would be less than 2 and speaking prophecies. Yes, maybe claim that a 1 and 1/2 year old girl could say anything back then? But 5-6 years old makes more sense.
    – David Kenner
    Sep 17, 2018 at 21:17
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    @SAH Aharon was 3 years older than Moshe (Shmos 7:7). Miriam was older than Moshe too ( She spoke with Bas-Pharoah when Moshe was 3 months old. Shmos 2:1-8). So she is either in between the age of Moshe and Aharon or older than both of them. Now, if Miriam shortly before the conception of Moshe, encouraged her father Amram to remarry her mother Yocheved (by prophecy); that would make Miriam about 2 years old (?) (give or take "X" months), when she spoke to Amram about remarrying her mother (see Rashi to Shmos 2:1). So it makes more sense that she was a few years older than both.
    – David Kenner
    Sep 18, 2018 at 8:28
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There's another Ramban which insinuates that Miriam was older than Aaron (shemot 2:1) at the end:

ואע"פ שהיה אהרן קטן, נתן השם שמחה בלבו בענין זה, או מרים מלמדתו.

The Gemara Sota says the 2 children danced at their parents' second wedding, and the Ramban suggests that (the older) Miriam taught Aaron how to rejoice.

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Another data point, also showing that Miriam was older:

Shir Hashirim Rabbah 2:11 says that the most intense period of the Jews' enslavement in Egypt began when Miriam was born (a note inserted into the text says that this is why she was named Miriam, related to maror, bitter), and lasted for 86 years.

Shemos 7:7 records that Aharon was 83 years old when he and Moshe confronted Pharaoh, which was shortly before the Exodus.

So at the time of the Exodus, Aharon was 83 or 84 years old, and Miriam was 86 or 87.

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