Depends on the gift...
If it's a cute personalised hooded bath towel - the name she gets called by now
If it's a drinking bottle/cup - same
If it's one of those books where the story is written with the main character as the child - same
For all of the above, if the child is called by/used to using Betty, then putting Elizabeth on it would be odd.
What does she get introduced as? (as in 'here's baby xxxx'). If the answer is Betty, then definitely put Betty.
If it's something they'll keep into older childhood/young adulthood then maybe use their full name. But it's harder to know, as you don't know if she'll stay as Betty, or maybe adopt Elizabeth as her name.
Changed the name but the scenario is true here - my full name is Victoria, but from birth my parents called me Tor, I grew up as Tor and friends knew me as Tor. If I'd been given something with Victoria on it, it would have been very odd.
When I got married, Tor didn't sound right with my new surname (and I was moving to a different part of the country, so meeting a whole load of new people) so I 'became' Victoria from then onwards. I don't have much that's personalised now, but for anything that is (race t-shirts at marathons!) I use Victoria, as that's what I go by now. But family still call me Tor, as do 'old' friends.