Motivational Desire
As much as the above theories may be true (and I agree with him making it up), why has it not come unto anyone's thought to just say... Maybe he can flat out ask his TARDIS? It has everything it'd need to, and it's sentient at that. Part of the time vortex or whatever... So phone the Blue Box.
Also, to those mentioning the difference in time spans on a planet. The series has made it clear that by the year 5000 (or whenever the hospital came active), the universe had a standard in calculations and warnings in how it'd work. So the difference between a Gallifreyan Year and Earth Year would be the same by that point and he'd have an accurate figure to relate to.
I like the 'ask-the-TARDIS' solution. It sounds about plausible. xD
I mean, yeah, the TARDIS would have a standardized measurement of time and what not, so even if as time-travelers, they didn't go through time the same way as non-time travelers since you could theoretically, spend 24 hours in several 'days', all the TARDIS can really count is how long the Doctor himself has been around and then caluculate it in to the equivalent amount of time.
Tho tbh, it still makes me wonder if the TARDIS could do that if the Doctor was isolated from it for an extended amount of time. I mean, upon being reunited, how accurately would the TARDIS be able to say how long the Doctor spent away from her to be able to accurately know his age?
Does what I'm saying even make sense? LOL I don't even know myself. xD