If someone has his bar mitzva during s'firas haomer and said the bracha every night so far, can he still make a bracha? (I will be asking a Rav for the actual P'sak Halacha)
One source that discusses this is Minchas Chinuch (mitzvah 306). He seems to end up saying that the boy can indeed continue counting with a berachah, since his earlier Rabbinic obligation (because of chinuch, education in the performance of mitzvos) counts towards his new Biblical one.
A related issue is the fact that according to many opinions, the mitzvah of counting the Omer nowadays (in the absence of the Beis Hamikdash and the Omer offering) is only a Rabbinic obligation (Beis Yosef, Orach Chaim 489). In one of his talks, the Lubavitcher Rebbe zt"l observes that if Moshiach comes during the Sefirah period (and may this indeed happen!), then according to Minchas Chinuch's logic we will indeed be able to continue counting with a berachah, for the same reason.
Sepharadim (following the psak of Rav Ovadia Yosef, see Yalkut Yosef siman 489 seif 10) rules that someone who becomes bar mitzvah during the omer may not continue to count with a beracha, even if he had been careful to count every day leading up to then. Sephardim are very careful about the principle of safek berachot l'hakel and in almost any circumstance where there's a credible doubt in the poskim about saying a beracha, they will omit the beracha. (Ashkenazim may rule differently, but I don't have any appropriate seforim to consult.)
I heard a shiur on this question which included many approaches of which some are mentioned in previous answers. One novel approach, though I cannot remember the source:
The idea of temimos, having a complete counting, does not mean that sefirah is one big counting. If it were, we would not make a bracha every night, but at the beginning. Rather, each day is a mitzva, conditional that there was no lapse in his counting.
This is where my memory is fuzzy- I believe the upshot was that he may count with a bracha because there was no lack in his counting since he had been counting albeit without a mitzva.