The non-canon answer to the OP's question is appended at the bottom just for information's sake, but the main body will deal with what the current Disney canon says.
The question is: When and why did the Rebel Alliance adopt the phrase "May the Force be with you"?.
In the new canon I could not find an answer specifically to the "When" part, but the canon does suggest some explanation for the "Why" part, or at least why it is not unreasonable for the Rebel Alliance to use this phrase and/because of the connection with the Jedi that goes along with it.
In Star Wars Rebels, the only two instances of the phrase being used are:
After this, we next hear it in ANH (in 0 BBY) when General Jan Dodonna told the Rebel pilots:
Then man your ships, and may the Force be with you.
which is what the OP is referencing.
I mention SWR, which is ongoing, because the Lothal rebels come into contact with Bail Organa, Ahsoka Tano and Jun Sato, all of whom are leaders of the as-yet-fledgling Rebellion, and none of them have used the phrase by the end of Season Two of the series.
The case for why the Rebel Alliance uses the phrase, apart from when it started doing so (unless that becomes relevant, see N.B. below), is suggested by the history of the Alliance/Rebellion itself, its leaders, and its precursor which is referred to as 'the rebellion'.
Before the Alliance to Restore the Republic, there had been a growing opposition to the Galactic Empire that had metastasized into various incidents (such as the rebellion on Ryloth) and eventually into a series of rebel cells, two of which are depicted in SWR (the Phoenix rebel cell and the aforementioned Lothal rebels). This growing rebel movement was coordinated by Senator Bail Organa, Senator Mon Mothma, and former Jedi Padawan Ahsoka Tano.
By the time of the Battle of Yavin and the events of ANH, this organisation had transformed into the Alliance to Restore the Republic (or Rebel Alliance), which was headed by Bail Organa (before his death on Alderaan), Mon Mothma and Admiral Ackbar, as well as Leia Organa.
Connection with the Jedi
Even though the general public may not have been generally aware of the Jedi Order, or were not inclined to associate with it during the time of the Galactic Republic (which is discussed somewhat here), the leaders of the Rebel Alliance were certainly aware of the Jedi Order:
Mon Mothma had served in the Galactic Senate during the Clone Wars and was a very good friend of Padme Amidala
Admiral Ackbar had served alongside the Jedi when they helped to liberate his planet, Mon Cala, from a CIS-manufactured civil war, as depicted in TCW 4x01-02
Bail Organa served very closely with the Jedi during the Clone Wars, and was friends with Mon Mothma and Padme Amidala, and had participated in several of the Jedi's adventures as shown in TCW. In SWR, Organa reveals himself as a leader of the rebel movement which includes former Jedi Padawan Ahsoka Tano whom he knew from TCW. Also, at the end of ROTS, when he and Yoda and Obi-wan Kenobi are splitting up the Skywalker kids, Yoda plants the seeds of the Rebellion when he says:
Until the time is right, disappear we will
- Leia Organa worked with her father in the Rebellion and evidenced her knowledge of the Jedi at least in regards to knowing of Obi-wan Kenobi (as a general in the Clone Wars) in ANH as discussed here, here and here
There is no evidence that the Rebel Alliance used the phrase outside of their organisation, so it did not necessarily have to resonate with the public at large, only internally.
What's in a Name?
The Rebel Alliance's full name, the Alliance to Restore the Republic also indicates a possible motivation. At the end of the Clone Wars, the Jedi discovered the identity of Chancellor Palpatine as Darth Sidious and confronted him. After Palpatine survived this, he accused the Jedi of having betrayed him, the Senate, and the people of the Republic, and used this as his justification, publicly, for both eradicating the Jedi Order, and for the necessity of the security measures which led to the creation of the Galactic Empire.
The Alliance was an organisation which was dedicated to restoring the Republic, and its name indicates that it was rejecting not just the Empire, but also its foundations and 'legitimacy', This is particularly so since at least one of its founders, Bail Organa, knew very well that the story Palpatine had told the public was a lie. Organa almost certainly would have known the role that Anakin Skywalker had played (even if the general public did not as discussed here and here), and that he had become Darth Vader since both Kenobi and Yoda discovered this near the end of ROTS (as discussed here) and before their strategy meeting with Organa.
It is a possibility therefore that they wished to actively associate themselves with the Jedi, knowing the antagonism this would have for the Empire, especially as they were aware of the true history of the Clone Wars and the creation of the Empire that they opposed, in that context the phrase would resonate with them and possess true meaning in the light of their struggle.
In addition, at least one of the crew members of the Lothal rebels, who apparently go on to become the core of the Rebellion, had a somewhat direct connection with the Jedi. Hera Syndulla was the daughter of Cham Syndulla, who fought alongside the Jedi (including Ahsoka Tano) during the Clone Wars to free Ryloth from the CIS, twice, as shown in TCW 1x21 and TCW 3x03 (which features Bail Organa also helping). Cham Syndulla was the leader of the Free Ryloth Movement referenced above.
The Jedi were directly and indirectly involved in the rebel movement from the beginning to the end. By the time of the Battle of Yavin, when General Jan Dodonna uses the phrase infront of Luke Skywalker;
former Jedi had been directly involved in the movement for up to 5 years, having played a key role in organising its early origins (Ahsoka Tano), and running some of its early missions (Kanan Jarrus, who at the time is the only other Jedi to be shown involved in the movement)
the Jedi could be considered the nexus for the entire enterprise starting in the events of the Clone Wars and the birth of the Galactic Empire, and therefore could be considered an internally-motivating connection for the resistance
the Jedi may have played an as-of-yet unknown role in it leading to up to that point (see N.B.)
As the phrase itself is a symbol of encouragement, a direct challenge to the Galactic Empire (as the OP states), a rejection of the very legitimacy itself of that Empire, a harkening back to the Republic they wished to restore, and a reference to the key role that the Jedi have played in its origins and (early) operations, an argument can be made that the phrase was meaningfully chosen.
The non-canon answer is found in the Force Unleashed video games. The Rebel Alliance coalesces around the events of Galen Marek and former Jedi Rahm Kota. These two Jedi, along with Bail Organa, Mon Mothma, Leia Organa and others, are involved in the creation of the Rebel Alliance. For the role that Galen Marek played in ultimately giving birth to, and protecting, this movement, the rebels chose the insignia of his house to be the seal for their movement. In this 'timeline' the Jedi are directly responsible for the creation of the Rebellion (the Empire attempting to draw out its enemies, showing them the Death Star when they were imprisoned on it, the inspirational role that Galen Marek played) and so the phrase has significance in this context as well.
Although TFU is no longer considered canon, it is unclear if any remnant of this remains in the canon, since the Rebel Alliance still uses this seal, and the Resistance in TFA uses an orange version of the same symbol as well. This may be retconned later since in SWR, Sabine Wren uses a version of the starbird as a personal signature, and that particular theory is discussed off-site here. A general discussion about the starbird logo can be found here
From ANH, TESB, ROTJ and TFA, we see that there is no mention of either Ahsoka Tano, Kanan Jarrus or Ezra Bridger, all key Jedi figures in the Rebellion, suggesting that they are either dead, or just off-camera. Mirroring the recognition nature of the seal as discussed in the non-canon answer above, it's possible that the resistance may commemorate the role of the Jedi in the Rebellion by using this phrase, but that is entirely speculation since the events that this would refer to have not yet been depicted, we don't yet know the fates of Ahsoka, Kanan or Ezra or what other role the Jedi may have played in the Rebellion. Other ideas are discussed off-site here.