Her Stuartness didn't give her a better claim per se although she might see it as a better other side of the family than Elizabeth's Boleyn connection (less royal than Mary). The main thing that might make her and others go so far as to say that Mary's claim was better was because of the religious issues of Anne Boleyn's marriage however.

It is pretty public knowledge than Catherine of Aragon was still alive while Henry married Anne, and that she defended the validity of her marriage. Meanwhile Henry marrying Anne was based on it being invalid. We'll never really know the truth, and IMO it doesn't really matter because by our standards they should have divorced (there was no way the relationship was going to work by that stage and Catherine was mainly avoiding the scandal associated with divorce at the time, as well as defending her own honesty on the virginity point). But the point is at the time marriage law was very strict and very religiously based and her Hapsburg relatives took this very seriously. So from Mary's point of view having been the daughter of a Catholic Scottish king and marrying a Catholic french heir she was firmly on the Hapsburg side of that argument, which also fit because Scotland and France were maintaining the Auld Alliance against England. The Protestant Lords of the Congregation broke that alliance and rebelled against Mary's mother Mary of Guise, but Mary clearly still favoured it, albeit while diplomatic with England. It's significant to note that technically Elizabeth was briefly declared a bastard by Henry for a few years between Anne's execution and when she was brought back into the line of succession, without technically revoking the earlier decree (because it included Anne's condemnation which Henry wanted to keep). So there was a kind of legal argument, albeit the previous monarch and Elizabeth undisputed father put her in the line of succession and she had waited per those terms until after her brother and sister.

Mary's claim came from her grandmother Margaret Tudor, the eldest daughter of Henry VII, the first Tudor king, and Elizabeth of York, daughter of Yorkist king Edward IV. She was married to James IV who in turn was father to James V, Mary's mother. Those Stuarts with Margaret's blood, and her later children by Archibald Douglas, technically had the strongest blood claim after Henry's children. However, Henry's will favoured the Suffolk line over them i.e. daughters of his favoured and younger sister Mary by his favourite the Duke of Suffolk. And there were two Grey sisters during Elizabeth's time that had that claim, technically superior to Mary's according to Henry's will. This was the central problem with succession disputes, especially post-reformation where people were ruling each other's marriages invalid; there was no real system to sort out claims when wills clashed with blood disputes. If you believed someone, e.g. Elizabeth was illegitimate you could rule them out at least for those who believed it, but how can you validly sort between the Stuarts and the Greys? In the end blood won out when the Grey sisters were disgraced, and probably had no designs on the throne, and Mary's Protestant son James acceded as James I of England, which was an acceptable compromise for enough people.

Publicly at least, Mary's declared intention was only to gain recognition for her claim to be Elizabeth's official successor. When she was in France, she was declared Queen of England and arms were displayed, but that was decided by the King of France, the Dauphin's father and she was too young to assume she supported that merely because she went along with it. She also did refuse to affirm the Treaty of Edinburgh, but she was still only seventeen, in France and her mother had just died. Her motives probably had more to do with the removal of the French troops supporting Mary and the fact it was encouraged by the Lords of Congregation, who had rebelled against her mother as regent. Perhaps she did also think Elizabeth was illegitimate however. One cannot say for sure. Supporters like Anthony Babington, the Spanish Hapsburgs, her Guise relatives and Charles Neville, 6th Earl of Westmorland definitely did support Elizabeth actually being deposed in favour of Mary. However, the evidence seems to indicate that Mary only really supported a plot when it finally came to Babington and even then her actions seem to have shown some reservation (some people argue the Beer-Barrell post used; where the messenger was a spy sent to entice her, counts as a sting operation, see John Guy).

TLDR; Elizabeth's validity as a royal heir was contested because of Anne's contested marriage. Mary was a Catholic so she was raised on the other side of the argument. Theoretically, she might have thought she was a more legitimate claimant than Elizabeth therefore. However, I'm not aware of any evidence she said so, only that she was the best successor to Elizabeth, which Elizabeth I herself agreed with. If Elizabeth was illegitimate, Henry's will and Margaret's senior line means there were competing claims between the Seymour/Grey family and the Stuarts, so even then it was not clear.