Valeyard is one of the classic eras' most convoluted villains, and the revival series of Doctor Who has thus far ignored him, barring brief mentions in "The Name of the Doctor" and "Twice Upon a Time." However, if Tennant is returning to the series, this bizarre bad guy could be the perfect role for the veteran actor, and it could also solve one of the series' biggest continuity errors.
The Valeyard is a confusing Doctor Who villain who is a foe of the Sixth Doctor (Colin Baker). According to the Master, the Valeyard is an "amalgamation of the darker side of [the Doctor's] nature, taken somewhere between [their] 12th and final incarnation." Prior to the events of "The Time of the Doctor" and "The Timeless Children" rewriting the Doctor's regeneration limit, this would put the Valeyard as an offshoot of the 11th and 12th Doctor (as the presence of the War Doctor bumps the numbers up by one). However, the modern series has yet to address this plot point.
David Tennant's reappearance on Doctor Who is the perfect time to bring back the Valeyard. It is already unprecedented for an actor to play the Doctor twice, so showrunner Russell T. Davies could have some fun with the actor's return and slowly reveal him as a villainous version of the Doctor. Furthermore, Tennant has proved how excellent he is at playing villainous characters with turns as Kilgrave in Harry Potter and the Goblet of Fire.
Having Tennant play a rogue offshoot of the Doctor's next regeneration (without the audience knowing at first) could work with the existing continuity, albeit with a few "timey-wimey" tweaks. It has been established that the Doctor is actually a being known as the Timeless Child, meaning that they have at least seven regenerations before the "First" Doctor, originally portrayed by William Hartnell. This means that all semblance of the Valeyard being between the 11th and 12th Doctor has been made redundant, and Davies could easily write around it.
Bringing the Valeyard back would also represent a major moment for classic Doctor Who audiences, as his identity and where he fits into canon has always been a point of debate. Davies could bring Tennant back and finally put this mystery to bed, perhaps with the 13th Doctor (Jodie Whittaker) undergoing a corrupted regeneration, creating both the 14th Doctor and the Valeyard. The showrunner could come up with an interesting mystery as to why the character looks like the 10th Doctor, and Peter Capaldi's 12th Doctor proves that there is precedent for the Doctor choosing their new face to some extent.
If Doctor Who really is bringing David Tenannt back to the show, it would be an incredible twist to see him return as a dark variant of the Doctor. If that is the way the BBC wants to use him, then finally solving the mystery of the Valeyard and bringing the character into the revival era would be too good an opportunity to miss. David Tennant is a beloved part of the series, and he's played compelling villains before, so it's a match made in heaven.